View source for Nicolas Bourbaki

{{Short description|Pseudonym of a group of mathematicians}}
{{About|the group of mathematicians|the family of French officers|Bourbaki family (disambiguation)|the computer scientist|Nikolaos Bourbakis }}
{{Infobox organization
| name                = Association of Collaborators of Nicolas Bourbaki
| purpose             = Publication of textbooks in [[pure mathematics]]
| headquarters        = [[École normale supérieure (Paris)|École Normale Supérieure]], Paris
| type                = [[Voluntary association]]
| language            = [[French language|French]]
| named_after         = [[Charles-Denis Bourbaki]]
| formation           = 10 December 1934 (first unofficial meeting)<br />10–17 July 1935 (first official, founding conference)
| founding_location   = [[Latin Quarter]], Paris, France (first unofficial meeting)<br />[[Besse-en-Chandesse]], France (first official, founding conference)
| founders            = {{unbulleted list| [[Henri Cartan]] | [[Claude Chevalley]] | [[Jean Coulomb]] | [[Jean Delsarte]] | [[Jean Dieudonné]] | [[Charles Ehresmann]] | [[Szolem Mandelbrojt]] | [[René de Possel]]| [[André Weil]] }}
| membership          = Confidential
| website             = {{URL|http://www.bourbaki.fr/}}
| native_name         = Association des collaborateurs de Nicolas Bourbaki
| formerly            = Committee for the Treatise on Analysis
| image               = Bourbaki congress1938.png
| caption             = Bourbaki congress at [[Dieulefit]] in 1938. From left, [[Simone Weil]],{{efn|Simone Weil was not a member of the group; she was a philosopher, not a mathematician.  However she attended multiple early conferences to support her brother André, and also to learn mathematics.{{sfn|Aczel|pp=123–25}}}} [[Charles Pisot]], [[André Weil]], [[Jean Dieudonné]] (sitting), [[Claude Chabauty]], [[Charles Ehresmann]], and [[Jean Delsarte]].{{sfn|Mashaal|p=31}}
}}

'''Nicolas Bourbaki''' ({{IPA|fr|nikola buʁbaki|lang}}) is the collective [[pseudonym]] of a group of mathematicians, predominantly French alumni of the [[École normale supérieure (Paris)|École normale supérieure]] (ENS). Founded in 1934–1935, the '''Bourbaki group''' originally intended to prepare a new [[textbook]] in [[mathematical analysis|analysis]]. Over time the project became much more ambitious, growing into a large series of textbooks published under the Bourbaki name, meant to treat modern [[pure mathematics]]. The series is known collectively as the ''[[Éléments de mathématique]]'' (''Elements of Mathematics''), the group's central work. Topics treated in the series include [[set theory]], [[abstract algebra]], [[topology]], analysis, [[Lie group]]s and [[Lie algebra]]s.

Bourbaki was founded in response to the effects of the [[First World War]] which caused the death of a generation of French mathematicians; as a result, young university instructors were forced to use dated texts. While teaching at the [[University of Strasbourg]], [[Henri Cartan]] complained to his colleague [[André Weil]] of the inadequacy of available course material, which prompted Weil to propose a meeting with others in Paris to collectively write a modern analysis textbook. The group's core founders were Cartan, [[Claude Chevalley]], [[Jean Delsarte]], [[Jean Dieudonné]] and Weil; others participated briefly during the group's early years, and membership has changed gradually over time. Although former members openly discuss their past involvement with the group, Bourbaki has a custom of keeping its current membership secret.

The group's name derives from the 19th century French general [[Charles-Denis Bourbaki]], who had a career of successful military campaigns before suffering a dramatic loss in the [[Franco-Prussian War]].<ref name=":0">{{Cite book|title=The Apprenticeship of a Mathematician|url=https://archive.org/details/apprenticeshipma00weil|url-access=limited|last=Weil|first=André|publisher=Birkhäuser Verlag|year=1992|isbn=978-3764326500|pages=[https://archive.org/details/apprenticeshipma00weil/page/n91 93–122]|author-link=André Weil}}</ref> The name was therefore familiar to early 20th-century French students. Weil remembered an ENS [[student prank]] in which an upperclassman posed as a professor and presented a "theorem of Bourbaki"; the name was later adopted.

The Bourbaki group holds regular private conferences for the purpose of drafting and expanding the ''Éléments''. Topics are assigned to subcommittees, drafts are debated, and unanimous agreement is required before a text is deemed fit for publication. Although slow and labor-intensive, the process results in a work which meets the group's standards for [[rigour]] and generality. The group is also associated with the [[Séminaire Bourbaki]], a regular series of lectures presented by members and non-members of the group, also published and disseminated as written documents. Bourbaki maintains an office at the ENS.{{sfn|Beaulieu|1999|p=221}}

Nicolas Bourbaki was influential in 20th-century mathematics, particularly during the middle of the century when volumes of the ''Éléments'' appeared frequently. The group is noted among mathematicians for its rigorous presentation and for introducing the notion of a [[mathematical structure]], an idea related to the broader, interdisciplinary concept of [[structuralism]].{{sfn|Aczel|pp=129–48}}{{sfn|Aubin|p=314}}{{sfn|Mashaal|pp=70–85}} Bourbaki's work informed the [[New Math]], a trend in elementary math education during the 1960s. Although the group remains active, its influence is considered to have declined due to infrequent publication of new volumes of the ''Éléments''. However, since 2012 the group has published four new (or significantly revised) volumes, the most recent in 2023 (treating [[spectral theory]]). Moreover, at least three further volumes are under preparation.

==Background==
[[File:Portrait Bourbaki.JPG|thumb|left|200px|[[Charles-Denis Bourbaki]], 19th century general and namesake of the collective]]

[[Charles-Denis Bourbaki|Charles-Denis Sauter Bourbaki]] was a successful general during the era of [[Napoleon III]], serving in the [[Crimean War]] and other conflicts. During the [[Franco-Prussian war]] however, Charles-Denis Bourbaki suffered a major defeat in which the [[Armée de l'Est]], under his command, retreated across the Swiss border and was disarmed. The general unsuccessfully attempted suicide. The dramatic story of his defeat was known in the French popular consciousness following his death.{{sfn|Aczel|pp=61–63}}{{sfn|Mashaal|pp=22–25}}

[[File:Gustav_Herglotz,_Gaston_Julia.jpeg|thumb|right|250px|[[Gaston Julia]] (right), who was not a member of Bourbaki, lost his nose during World War I. The war created a lost generation of mathematical knowledge, which the Bourbaki founders sought to fill.]]  In the early 20th century, the First World War affected Europeans of all professions and social classes, including mathematicians and male students who fought and died in the front. For example, the French mathematician [[Gaston Julia]], a pioneer in the study of [[fractal]]s, lost his nose during the war and wore a leather strap over the affected part of his face for the rest of his life. The deaths of ENS students resulted in a [[lost generation]] in the French mathematical community;{{sfn|Borel|p=373}} the estimated proportion of ENS mathematics students (and French students generally) who died in the war ranges from one-quarter to one-half, depending on the intervals of time (c. 1900–1918, especially 1910–1916) and populations considered.{{sfn|Aczel|p=82}}{{sfn|Mashaal|pp=44–45}} Furthermore, Bourbaki founder [[André Weil]] remarked in his memoir ''Apprenticeship of a Mathematician'' that France and Germany took different approaches with their intelligentsia during the war: while Germany protected its young students and scientists, France instead committed them to the front, owing to the French [[Liberté, égalité, fraternité|culture]] of [[egalitarianism]].{{sfn|Mashaal|pp=44–45}}

A succeeding generation of mathematics students attended the ENS during the 1920s, including Weil and others, the future founders of Bourbaki. During his time as a student, Weil recalled a prank in which an upperclassman, {{ill|Raoul Husson|fr|Raoul Husson}}, posed as a professor and gave a math lecture, ending with a prompt: "Theorem of Bourbaki: you are to prove the following...". Weil was also aware of a similar stunt around 1910<ref name=":0" /> in which a student claimed to be from the fictional, impoverished nation of "Poldevia" and solicited the public for donations.{{sfn|Aczel|pp=63–65}}{{sfn|Mashaal|p=23}} Weil had strong interests in languages and [[Indian culture]], having learned [[Sanskrit]] and read the [[Bhagavad Gita]].{{sfn|Aczel|pp=25–26}}{{sfn|Mashaal|pp=35–37}} After graduating from the ENS and obtaining his doctorate, Weil took a teaching stint at the [[Aligarh Muslim University]] in India. While there, Weil met the mathematician [[Damodar Dharmananda Kosambi|Damodar Kosambi]], who was engaged in a power struggle with one of his colleagues. Weil suggested that Kosambi write an article with material attributed to one "Bourbaki", in order to show off his knowledge to the colleague.{{sfn|Beaulieu|1999|p=239}} Kosambi took the suggestion, attributing the material discussed in the article to "the little-known Russian mathematician '''D. Bourbaki''', who was poisoned during the Revolution." It was the first article in the mathematical literature with material attributed to the eponymous "Bourbaki".{{sfn|Aczel|p=65}}<ref>{{cite book |chapter=On a Generalization of the Second Theorem of Bourbaki |last=Kosambi |first=Damodar Dharmananda |title=D.D. Kosambi |pages=55–57 |doi=10.1007/978-81-322-3676-4_6 |isbn=978-81-322-3674-0 |year=2016 }}</ref>{{sfn|Mashaal|p=26}} Weil's stay in India was short-lived; he attempted to revamp the mathematics department at Aligarh, without success.{{sfn|Mashaal|p=35}} The university administration planned to fire Weil and promote his colleague [[Tirukkannapuram Vijayaraghavan|Vijayaraghavan]] to the vacated position. However, Weil and Vijayaraghavan respected one another. Rather than play any role in the drama, Vijayaraghavan instead resigned, later informing Weil of the plan.{{sfn|Aczel|pp=32–34}} Weil returned to Europe to seek another teaching position. He ended up at the University of Strasbourg, joining his friend and colleague Henri Cartan.{{sfn|Aczel|p=81}}

==The Bourbaki collective==

[[File:Integral as region under curve.svg|thumb|left|200px|Bourbaki was founded to produce a text in [[mathematical analysis]], a branch of mathematics entailing [[calculus]]]]

===Founding===
During their time together at Strasbourg, Weil and Cartan regularly complained to each other regarding the inadequacy of available course material for [[calculus]] instruction. In his memoir ''Apprenticeship'', Weil described his solution in the following terms: "One winter day toward the end of 1934, I came upon a great idea that would put an end to these ceaseless interrogations by my comrade. 'We are five or six friends', I told him some time later, 'who are in charge of the same mathematics curriculum at various universities. Let us all come together and regulate these matters once and for all, and after this, I shall be delivered of these questions.' I was unaware of the fact that Bourbaki was born at that instant."{{sfn|Aczel|p=81}} Cartan confirmed the account.{{sfn|Mashaal|p=4}}

The first, unofficial meeting of the Bourbaki collective took place at noon on Monday, 10 December 1934, at the Café Grill-Room A. Capoulade, Paris, in the [[Latin Quarter]].{{sfn|Aczel|pp=82–83}}{{sfn|Beaulieu|1993|p=28}}{{sfn|Mashaal|p=6}}<ref name="mactutor">{{cite web |url=http://mathshistory.st-andrews.ac.uk/HistTopics/Bourbaki_1.html |title=Bourbaki: the pre-war years |last1=O'Connor |first1=John J. |last2=Robertson |first2=Edmund F. |date=December 2005 |website=[[MacTutor History of Mathematics archive|Mactutor]]}}</ref>{{efn|The restaurant, which no longer exists, was at 63 Boulevard Saint-Michel.{{sfn|Beaulieu|1993|p=29}} }} Six mathematicians were present: [[Henri Cartan]], [[Claude Chevalley]], [[Jean Delsarte]], [[Jean Dieudonné]], [[René de Possel]], and [[André Weil]]. Most of the group were based outside Paris and were in town to attend the Julia Seminar, a conference prepared with the help of Gaston Julia at which several future Bourbaki members and associates presented.{{sfn|Beaulieu|1993|p=32}}{{sfn|Mashaal|pp=6–7, 102–03}}{{efn|The Julia Seminar was held every other Monday, in the afternoon.{{sfn|Mashaal|p=103}}  Bourbaki's early lunch meetings during 1934–1935 were typically held on the same Mondays, immediately before the Seminar.{{sfn|Beaulieu|1993|p=32}}<ref name="barchive">{{cite web |url=http://sites.mathdoc.fr/archives-bourbaki/ |title=Archives de l'Association des Collaborateurs de Nicolas Bourbaki}}</ref><ref name="timedate">{{cite web |url=https://www.timeanddate.com/calendar/?year=1935&country=5 |title=Calendar for Year 1935 (France) |website=Time and Date}}</ref> }} The group resolved to collectively write a treatise on analysis, for the purpose of standardizing calculus instruction in French universities. The project was especially meant to supersede the text of [[Édouard Goursat]], which the group found to be badly outdated, and to improve its treatment of [[Stokes' Theorem]].{{sfn|Mashaal|p=6}}{{sfn|Aczel|p=84}}{{sfn|Beaulieu|1999|p=233}}<ref name="numericana">{{cite web |url=http://www.numericana.com/fame/bourbaki.htm |title=The Many Faces of Nicolas Bourbaki |last=Michon |first=Gérard P. |website=Numericana}}</ref> The founders were also motivated by a desire to incorporate ideas from the [[University of Göttingen|Göttingen]] school, particularly from exponents [[David Hilbert|Hilbert]], [[Emmy Noether|Noether]] and [[Bartel van der Waerden|B.L. van der Waerden]]. Further, in the aftermath of World War I, there was a certain nationalist impulse to save French mathematics from decline, especially in competition with Germany. As Dieudonné stated in an interview, "Without meaning to boast, I can say that it was Bourbaki that saved French mathematics from extinction."{{sfn|Mashaal|pp=38-45}}

Jean Delsarte was particularly favorable to the collective aspect of the proposed project, observing that such a working style could insulate the group's work against potential later individual claims of [[copyright]].{{sfn|Aczel|p=84}}{{sfn|Mashaal|pp=7,14}}{{efn|Delsarte's favorable view of a collective project was not recorded in the minutes of the first meeting.  He is supposed to have expressed the view elsewhere, with Cartan and Weil eventually attributing the opinion to him.  However, the opinion is closely associated with the working style of Bourbaki which eventually emerged.{{sfn|Beaulieu|1993|pp=28–29}} }} As various topics were discussed, Delsarte also suggested that the work begin in the most abstract, axiomatic terms possible, treating all of mathematics prerequisite to analysis from scratch.{{sfn|Aczel|pp=85–86}}{{sfn|Aubin|p=303}} The group agreed to the idea, and this foundational area of the proposed work was referred to as the "Abstract Packet" (Paquet Abstrait).{{sfn|Aczel|p=86}}{{sfn|Beaulieu|1993|p=30}}{{sfn|Mashaal|p=11}} [[Working title]]s were adopted: the group styled itself as the '''Committee for the Treatise on Analysis''', and their proposed work was called the ''Treatise on Analysis'' (''Traité d'analyse'').{{sfn|Aczel|p=87}}{{sfn|Mashaal|p=8}} In all, the collective held ten preliminary biweekly meetings at A. Capoulade before its first official, founding conference in July 1935.{{sfn|Mashaal|p=8}}{{sfn|Beaulieu|1993|p=33}} During this early period, [[Paul Dubreil]], [[Jean Leray]] and [[Szolem Mandelbrojt]] joined and participated. Dubreil and Leray left the meetings before the following summer, and were respectively replaced by new participants [[Jean Coulomb]] and [[Charles Ehresmann]].{{sfn|Aczel|p=87}}{{sfn|Mashaal|pp=8–9}}

[[File:Nicolas Bourbaki naissance à Besse.jpg|thumb|left|Sign marking the official founding of Bourbaki in [[Besse-en-Chandesse]]]]

The group's official founding conference was held in [[Besse-en-Chandesse]], from 10 to 17 July 1935.{{sfn|Aczel|p=90}}{{sfn|Mashaal|p=10}} At the time of the official founding, the membership consisted of the six attendees at the first lunch of 10 December 1934, together with Coulomb, Ehresmann and Mandelbrojt. On 16 July, the members took a walk to alleviate the boredom of unproductive proceedings. During the malaise, some decided to [[Nude swimming|skinny-dip]] in the nearby [[Lac Pavin]], repeatedly yelling "Bourbaki!"{{sfn|Mashaal|p=22}} At the close of the first official conference, the group renamed itself "Bourbaki", in reference to the general and prank as recalled by Weil and others.{{sfn|Mashaal|p=11}}{{efn|The mathematician Sterling K. Berberian suggested another possible origin for the Bourbaki name: [[Octave Mirbeau|Octave Mirbeau's]] 1900 novel ''[[The Diary of a Chambermaid (novel)|The Diary of a Chambermaid]]'', which describes a hedgehog named Bourbaki that eats voraciously.  However Mashaal dismissed this connection as being unlikely since the founders never referred to the novel, but only to the general and the Husson anecdote.{{sfn|Mashaal|pp=25–26}} }} During 1935, the group also resolved to establish the mathematical [[personhood]] of their collective pseudonym by getting an article published under its name.{{sfn|Aczel|p=90}}{{sfn|Mashaal|pp=27–29}} A first name had to be decided; a full name was required for publication of any article. To this end, René de Possel's wife Eveline "baptized" the pseudonym with the first name of Nicolas, becoming Bourbaki's "godmother".{{sfn|Aczel|p=90}}<ref name="Mainard">{{Cite web|url=http://www.academie-stanislas.org/academiestanislas/images/Publications/TomeXVI/TomeXVI-Mainard01.pdf|title=Le Mouvement Bourbaki|last=Mainard|first=Robert|date=October 21, 2001|website=academie-stanislas.org}}</ref>{{sfn|Mashaal|p=27}}<ref name="mccleary">{{Cite web|url=http://www.math.vassar.edu/faculty/mccleary/Bourbaki.pdf|title=Bourbaki and Algebraic Topology|last=McCleary|first=John|date=December 10, 2004|website=math.vassar.edu|archive-url=https://web.archive.org/web/20061030215012/http://math.vassar.edu/faculty/McCleary/Bourbaki.pdf|archive-date=October 30, 2006}}</ref> This allowed for the publication of a second article with material attributed to Bourbaki, this time under "his" own name.<ref name="sur">{{cite journal |url=https://gallica.bnf.fr/ark:/12148/bpt6k31534/f1311.image |title=Sur un théorème de Carathéodory et la mesure dans les espaces topologiques |last=Bourbaki |first=Nicolas |journal=[[Comptes rendus de l'Académie des Sciences]] |volume=201 |pages=1309–11 |date=18 November 1935}}</ref> Henri Cartan's father [[Élie Cartan]], also a mathematician and supportive of the group, presented the article to the publishers, who accepted it.{{sfn|Mashaal|pp=27–29}}

At the time of Bourbaki's founding, René de Possel and his wife Eveline were in the process of divorcing. Eveline remarried to André Weil in 1937, and de Possel left the Bourbaki collective some time later. This sequence of events has caused speculation that de Possel left the group because of the remarriage,{{sfn|Mashaal|p=17}} however this suggestion has also been criticized as possibly historically inaccurate, since de Possel is supposed to have remained active in Bourbaki for years after André's marriage to Eveline.<ref>{{cite web |url=http://www.neverendingbooks.org/the-bumpy-road-to-the-first-bourbaki-congress |title=The Bumpy Road to the First Bourbaki Congress |date=22 October 2009 |website=neverendingbooks.org}}</ref>

===World War II===

Bourbaki's work slowed significantly during the [[Second World War]], though the group survived and later flourished. Some members of Bourbaki were Jewish and therefore forced to flee from certain parts of Europe at certain times. Weil, who was Jewish, spent the summer of 1939 in Finland with his wife Eveline, as guests of [[Lars Ahlfors]]. Due to their travel near the border, the couple were suspected as Soviet spies by Finnish authorities near the onset of the [[Winter War]], and André was later arrested.<ref>{{cite web |url=https://www.icmihistory.unito.it/portrait/nevanlinna.php |title=Rolf Nevanlinna |website=icmihistory.unito.it}}</ref> According to an anecdote, Weil was to have been executed but for the passing mention of his case to [[Rolf Nevanlinna]], who asked that Weil's sentence be commuted.{{sfn|Aczel|pp=17–36}} However, the accuracy of this detail is dubious.<ref>[[Osmo Pekonen]]: ''L'affaire Weil à Helsinki en 1939'', Gazette des mathématiciens 52 (avril 1992), pp. 13–20. With an afterword by André Weil.</ref> Weil reached the United States in 1941, later taking another teaching stint in [[São Paulo]] from 1945 to 1947 before settling at the [[University of Chicago]] from 1947 to 1958 and finally the [[Institute for Advanced Study]] in [[Princeton, New Jersey|Princeton]], where he spent the remainder of his career. Although Weil remained in touch with the Bourbaki collective and visited Europe and the group periodically following the war, his level of involvement with Bourbaki never returned to that at the time of founding.

Second-generation Bourbaki member [[Laurent Schwartz]] was also Jewish and found pickup work as a math teacher in rural [[Vichy France]]. Moving from village to village, Schwartz planned his movements in order to evade capture by the [[Nazi]]s.{{sfn|Senechal|pp=22–28}} On one occasion Schwartz found himself trapped overnight in a certain village, as his expected transportation home was unavailable. There were two inns in town: a comfortable, well-appointed one, and a very poor one with no heating and bad beds. Schwartz's instinct told him to stay at the poor inn; overnight, the Nazis raided the good inn, leaving the poor inn unchecked.{{sfn|Aczel|p=40}}

Meanwhile, Jean Delsarte, a Catholic, was mobilized in 1939 as the captain of an audio reconnaissance battery. He was forced to lead the unit's retreat from the northeastern part of France toward the south. While passing near the Swiss border, Delsarte overheard a soldier say "We are the army of Bourbaki";{{sfn|Aczel|p=98}}{{sfn|Mashaal|pp=20–24}} the 19th-century general's retreat was known to the French. Delsarte had coincidentally led a retreat similar to that of the collective's namesake.

===Postwar until the present===

{{multiple image
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Following the war, Bourbaki had solidified the plan of its work and settled into a productive routine. Bourbaki regularly published volumes of the ''Éléments'' during the 1950s and 1960s, and enjoyed its greatest influence during this period.{{sfn|Aczel|p=117}}{{sfn|Beaulieu|1999|p=237}} Over time the founding members gradually left the group, slowly being replaced with younger newcomers including [[Jean-Pierre Serre]] and [[Alexander Grothendieck]]. Serre, Grothendieck and Laurent Schwartz were awarded the [[Fields Medal]] during the postwar period, in 1954, 1966 and 1950 respectively. Later members [[Alain Connes]] and [[Jean-Christophe Yoccoz]] also received the Fields Medal, in 1982 and 1994 respectively.{{sfn|Mashaal|p=19}}

The later practice of accepting scientific awards contrasted with some of the founders' views.{{sfn|Guedj|p=19}} During the 1930s, Weil and Delsarte petitioned against a French national scientific "medal system" proposed by the [[Nobel prize|Nobel]] [[Nobel Prize in Physics|physics]] laureate [[Jean Perrin]]. Weil and Delsarte felt that the institution of such a system would increase unconstructive pettiness and jealousy in the scientific community.{{sfn|Mashaal|p=49}} Despite this, the Bourbaki group had previously successfully petitioned Perrin for a government [[Grant (money)|grant]] to support its normal operations.{{sfn|Mashaal|pp=14–16}} Like the founders, Grothendieck was also averse to awards, albeit for [[pacifism|pacifist]] reasons. Although Grothendieck was awarded the Fields Medal in 1966, he declined to attend the ceremony in Moscow, in protest of the Soviet government.<ref>{{cite web |url=http://www.icm2018.org/wp/2018/08/03/sir-michael-atiyah-shares-memory-of-fields-win/ |title=Sir Michael Atiyah shares memory of Fields win |date=3 August 2018 |publisher=[[International Congress of Mathematicians]] |access-date=15 February 2020 |archive-date=22 September 2019 |archive-url=https://web.archive.org/web/20190922045445/http://www.icm2018.org/wp/2018/08/03/sir-michael-atiyah-shares-memory-of-fields-win/ |url-status=dead }}</ref> In 1988, Grothendieck rejected the [[Crafoord Prize]] outright, citing no personal need to accept prize money, lack of recent relevant output, and general distrust of the scientific community.<ref name="crafoord">{{cite web |url=http://www.math.columbia.edu/~lipyan/CrafoordPrize.pdf |last=Grothendieck |first=Alexander |title=Crafoord Prize letter, English translation |access-date=2005-06-17 |url-status = bot: unknown|archive-url=https://web.archive.org/web/20060106062005/http://www.math.columbia.edu/~lipyan/CrafoordPrize.pdf |archive-date=6 January 2006 |df=dmy}}</ref>

Born to Jewish [[anarchist]] parentage, Grothendieck survived the [[Holocaust]] and advanced rapidly in the French mathematical community, despite poor education during the war.{{sfn|Aczel|pp=9–10}} Grothendieck's teachers included Bourbaki's founders, and so he joined the group. During Grothendieck's membership, Bourbaki reached an impasse concerning its foundational approach. Grothendieck advocated for a reformulation of the group's work using [[category theory]] as its theoretical basis, as opposed to set theory. The proposal was ultimately rejected{{sfn|Aubin|p=328}}{{sfn|Beaulieu|1999|pp=236–37}}{{sfn|Corry|2009|pp=38–51}} in part because the group had already committed itself to a rigid track of sequential presentation, with multiple already-published volumes. Following this, Grothendieck left Bourbaki "in anger".<ref name="numericana" />{{sfn|Senechal|pp=22–28}}{{sfn|Aczel|p=119}} Biographers of the collective have described Bourbaki's unwillingness to start over in terms of category theory as a missed opportunity.{{sfn|Senechal|pp=22–28}}{{sfn|Aczel|p=205}}{{sfn|Mashaal|pp=81–84}} However, Bourbaki has in 2023 announced that a book on category theory is currently under preparation (see below the last paragraph of this section).

During the founding period, the group chose the Parisian publisher [[Éditions Hermann|Hermann]] to issue installments of the ''Éléments''. Hermann was led by Enrique Freymann, a friend of the founders willing to publish the group's project, despite financial risk. During the 1970s, Bourbaki entered a protracted legal battle with Hermann over matters of copyright and [[royalty payment]]. Although the Bourbaki group won the suit and retained collective copyright of the ''Éléments'', the dispute slowed the group's productivity.{{sfn|Aczel|pp=205–206}}{{sfn|Mashaal|pp=7, 51–54}} Former member Pierre Cartier described the lawsuit as a [[pyrrhic victory]], saying: "As usual in legal battles, both parties lost and the lawyer got rich."{{sfn|Senechal|pp=22–28}} Later editions of the ''Éléments'' were published by [[Masson (publisher)|Masson]], and modern editions are published by [[Springer Science+Business Media|Springer]].<ref>[https://www.springer.com/series/47 Elements of Mathematics] series in Springer</ref> From the 1980s through the 2000s, Bourbaki published very infrequently, with the result that in 1998 ''[[Le Monde]]'' pronounced the collective "dead".{{sfn|Mashaal|p=146}}

However, in 2012 Bourbaki resumed the publication of the ''Éléments'' with a revised chapter 8 of algebra, the first 4 chapters of a new book on [[algebraic topology]], and two volumes on [[spectral theory]] (the first of which is an expanded and revised version of the edition of 1967 while the latter consist of three new chapters). Moreover, the text of the two latest volumes announces that books on [[category theory]] and [[modular forms]] are currently under preparation (in addition to the latter part of the book on algebraic topology).<ref>{{cite book |last1=Bourbaki |first1=Nicolas |title=Théories spectrales: Chapitres 1 et 2 — Seconde édition, refondue et augmentée|year=2019 |publisher=Springer |series=Éléments de mathématique |isbn=978-3030140632|page=II.299}}</ref><ref>{{cite book |last1=Bourbaki |first1=Nicolas |title=Théories spectrales: Chapitres 3 à 5 |year=2023b |publisher=Springer |series=Éléments de mathématique |isbn=978-3031195044|page=V.416}}</ref>

==Working method==

[[File:Finite coxeter.svg|thumb|right|400px|At [[Armand Borel]]'s urging, Bourbaki's treatment of Lie groups and Lie algebras included uncharacteristic illustrations, such as graphs of finite [[Coxeter systems]]<ref name="lie">{{cite book |last=Bourbaki |first=Nicolas |title=Lie Groups and Lie Algebras, Chapters 4–6 |publisher=Springer |date=2002 |isbn=978-3540691716 |pages=205–206}}</ref>]]

Bourbaki holds periodic conferences for the purpose of expanding the ''Éléments''; these conferences are the central activity of the group's working life. Subcommittees are assigned to write drafts on specific material, and the drafts are later presented, vigorously debated, and re-drafted at the conferences. Unanimous agreement is required before any material is deemed acceptable for publication.{{sfn|Aczel|p=92}}{{sfn|Borel|p=375}}{{sfn|Guedj|p=18}} A given piece of material may require six or more drafts over a period of several years, and some drafts are never developed into completed work.{{sfn|Borel|p=375}}{{sfn|Guedj|p=20}} Bourbaki's writing process has therefore been described as "[[Sisyphus|Sisyphean]]".{{sfn|Guedj|p=18}} Although the method is slow, it yields a final product which satisfies the group's standards for [[mathematical rigour]], one of Bourbaki's main priorities in the treatise. Bourbaki's emphasis on rigour was a reaction to the style of [[Henri Poincaré]], who stressed the importance of free-flowing mathematical [[intuition]] at the cost of thorough presentation.{{efn|"Bourbaki came to terms with Poincaré only after a long struggle. When I joined the group in the fifties it was not the fashion to value Poincaré at all. He was old-fashioned." —Pierre Cartier{{sfn|Senechal|pp=22–28}} }} During the project's early years, Dieudonné served as the group's scribe, authoring several final drafts which were ultimately published. For this purpose, Dieudonné adopted an impersonal [[writing style]] which was not his own, but which was used to craft material acceptable to the entire group.{{sfn|Aczel|p=116}}{{sfn|Borel|p=376}} Dieudonné reserved his personal style for his own work; like all members of Bourbaki, Dieudonné also published material under his own name,{{sfn|Mashaal|p=69}} including the nine-volume ''[[Treatise on analysis|Éléments d'analyse]]'', a work explicitly focused on analysis and of a piece with Bourbaki's initial intentions.

Most of the final drafts of Bourbaki's ''Éléments'' carefully avoided using illustrations, favoring a formal presentation based only in text and formulas. An exception to this was the treatment of Lie groups and Lie algebras (especially in chapters 4–6), which did make use of diagrams and illustrations. The inclusion of illustration in this part of the work was due to [[Armand Borel]]. Borel was minority-Swiss in a majority-French collective, and [[Self-deprecation|self-deprecated]] as "the Swiss peasant", explaining that [[visual learning]] was important to the Swiss national character.{{sfn|Senechal|pp=22–28}}{{sfn|Aczel|pp=111–12}} When asked about the dearth of illustration in the work, former member [[Pierre Cartier (mathematician)|Pierre Cartier]] replied:

{{blockquote|text=The Bourbaki were [[Puritan]]s, and Puritans are strongly opposed to pictorial representations of truths of their faith. The number of Protestants and Jews in the Bourbaki group was overwhelming. And you know that the [[Protestantism in France|French Protestant]]s especially are very close to Jews in spirit. |title=Pierre Cartier{{sfn|Senechal|pp=22–28}} }}

The conferences have historically been held at quiet rural areas.{{sfn|Beaulieu|1999|pp=225–26}} These locations contrast with the lively, sometimes heated debates which have occurred. Laurent Schwartz reported an episode in which Weil slapped Cartan on the head with a draft. The hotel's proprietor saw the incident and assumed that the group would split up, but according to Schwartz, "peace was restored within ten minutes."{{sfn|Mashaal|pp=112–13}} The historical, confrontational style of debate within Bourbaki has been partly attributed to Weil, who believed that new ideas have a better chance of being born in confrontation than in an orderly discussion.{{sfn|Borel|p=375}}{{sfn|Mashaal|pp=112–13}} Schwartz related another illustrative incident: Dieudonné was adamant that [[topological vector spaces]] must appear in the work before [[Integral|integration]], and whenever anyone suggested that the order be reversed, he would loudly threaten his resignation. This became an in-joke among the group; [[Roger Godement|Roger Godement's]] wife Sonia attended a conference, aware of the idea, and asked for proof. As Sonia arrived at a meeting, a member suggested that integration must appear before topological vector spaces, which triggered Dieudonné's usual reaction.{{sfn|Mashaal|pp=112–13}}

Despite the historical culture of heated argument, Bourbaki thrived during the middle of the twentieth century. Bourbaki's ability to sustain such a collective, critical approach has been described as "something unusual",<ref>{{cite book |last1=Sabelli |first1=Héctor |contribution=Foreword |contributor-last=Kauffman |contributor-first=Louis H. |contributor-link=Louis Kauffman |title=BIOS: a Study of Creation |year=2005 |publisher=[[World Scientific]] |location=Singapore |series=Series on Knots and Everything |volume=35 |page=423 |isbn=978-9812561039 }}</ref> surprising even its own members. In founder Henri Cartan's words, "That a final product can be obtained at all is a kind of miracle that none of us can explain."<ref name="Corry1997">{{cite journal |last=Corry |first=Leo |title=The Origins of Eternal Truth in Modern Mathematics: Hilbert to Bourbaki and Beyond |url=https://www.tau.ac.il/~corry/publications/articles/truth.html |journal=Science in Context |volume=10 |issue=2 |page=279 |date=1997 |doi=10.1017/S0269889700002659 |s2cid=54803469 }}</ref>{{sfn|Corry|2004|p=309}} It has been suggested that the group survived because its members believed strongly in the importance of their collective project, despite personal differences.{{sfn|Borel|p=375}}{{sfn|Aczel|p=115}} When the group overcame difficulties or developed an idea that they liked, they would sometimes say ''l'esprit a soufflé'' ("the spirit breathes").{{sfn|Borel|p=375}}{{sfn|Mashaal|p=112}} Historian Liliane Beaulieu noted that the "spirit"—which might be an [[avatar]], the group mentality in action, or Bourbaki "himself"—was part of an internal culture and mythology which the group used to form its identity and perform work.{{sfn|Beaulieu|1999|p=245}}

===Humor===

Humor has been an important aspect of the group's culture, beginning with Weil's memories of the student pranks involving "Bourbaki" and "Poldevia". For example, in 1939 the group released a wedding announcement for the marriage of "Betti Bourbaki" (daughter of Nicolas) to one "[[Hoist with his own petard|H. Pétard]]" (H. "Firecrackers" or "Hector Pétard"), a "lion hunter".{{sfn|Beaulieu|1999|pp=239–40}} Hector Pétard was itself a pseudonym, but not one originally coined by the Bourbaki members. The Pétard moniker was originated by [[Ralph P. Boas]], [[Frank Smithies]] and other [[Princeton]] mathematicians who were aware of the Bourbaki project; inspired by them, the Princeton mathematicians published an article on the "mathematics of lion hunting". After meeting Boas and Smithies, Weil composed the wedding announcement, which contained several mathematical puns.{{sfn|Mashaal|pp=30, 113–14}} Bourbaki's internal newsletter ''La Tribu'' has sometimes been issued with humorous subtitles to describe a given conference, such as "The Extraordinary Congress of Old Fogies" (where anyone older than 30 was considered a fogy) or "The Congress of the Motorization of the Trotting Ass" (an expression used to describe the routine unfolding of a mathematical proof, or process).{{sfn|Beaulieu|1999|p=226}}{{sfn|Mashaal|pp=110–11}}

During the 1940s–1950s,{{sfn|Beaulieu|1999|p=241}}{{sfn|Mashaal|pp=33–34}} the [[American Mathematical Society]] received applications for individual membership from Bourbaki. They were rebuffed by [[John Robert Kline|J.R. Kline]] who understood the entity to be a collective, inviting them to re-apply for institutional membership. In response, Bourbaki floated a rumor that Ralph Boas was not a real person, but a collective pseudonym of the editors of ''[[Mathematical Reviews]]'' with which Boas had been affiliated. The reason for targeting Boas was because he had known the group in its earlier days when they were less strict with secrecy, and he'd described them as a collective in an article for the ''[[Encyclopædia Britannica]]''.{{sfn|Aczel|pp=121–23}} In November 1968, a mock obituary of Nicolas Bourbaki was released during one of the seminars.{{sfn|Beaulieu|1999|pp=241–42}}<ref>{{cite web |url=http://www.neverendingbooks.org/according-to-groth-iv-22 |title=According to Groth. IV.22 |date=1 October 2016 |website=neverendingbooks.org |access-date=2018-10-24}}</ref>

The group developed some variants of the word "Bourbaki" for internal use. The noun "Bourbaki" might refer to the group proper or to an individual member, e.g. "André Weil was a Bourbaki." "Bourbakist" is sometimes used to refer to members<ref name="numericana" /> but also denotes associates, supporters, and enthusiasts.{{sfn|Beaulieu|1993|p=31}}{{sfn|Beaulieu|1999|p=227}} To "bourbakize" meant to take a poor existing text and to improve it through an editing process.{{sfn|Guedj|p=20}}

Bourbaki's culture of humor has been described as an important factor in the group's social cohesion and capacity to survive, smoothing over tensions of heated debate.{{sfn|Mashaal|p=115}} As of {{year}}, a [[Twitter]] account registered to "Betty_Bourbaki" provides regular updates on the group's activity.<ref name="betty">{{cite web |url=https://twitter.com/betty_bourbaki |title=Compte twitter officiel de l'Association des collaborateurs de N. Bourbaki |author=Betty_Bourbaki |website=Twitter}}</ref>

==Works==

Bourbaki's work includes a series of textbooks, a series of printed lecture notes, journal articles, and an internal newsletter. The textbook series ''Éléments de mathématique'' ''(Elements of mathematics)'' is the group's central work. The [[Séminaire Bourbaki]] is a lecture series held regularly under the group's auspices, and the talks given are also published as lecture notes. Journal articles have been published with authorship attributed to Bourbaki, and the group publishes an internal newsletter ''La Tribu'' (''The Tribe'') which is distributed to current and former members.<ref name="archiver">{{cite web |url=http://archives-bourbaki.ahp-numerique.fr/elements-mathematique |title=Éléments de Mathématique |website=Archives Bourbaki}}</ref>{{sfn|Mashaal|pp=108–09}}

===''Éléments de mathématique''===
{{main|Éléments de mathématique}}
{{Quote box
 | quote  = Like those before him, Bourbaki insisted on setting mathematics in a “formalized language” with crystal-clear deductions based on strict formal rules. When [[Bertrand Russell]] and [[Alfred North Whitehead]] applied this approach at the turn of the twentieth century, they famously filled over 700 pages with formal symbols before establishing the proposition usually abbreviated as [[1+1=2]]. Bourbaki's formalism would dwarf even this, requiring some 4.5 trillion symbols just to define the number [[1]].<ref>{{cite journal |last1=Mathias |first1=A. R. D. |title=A Term of Length 4 523 659 424 929 |journal=Synthese |date=2002 |volume=133 |issue=1/2 |pages=75–86 |doi=10.1023/A:1020827725055 |jstor=20117295 |url=https://www.jstor.org/stable/20117295 |access-date=5 January 2024 |issn=0039-7857 |quote=ABSTRACT. Bourbaki suggest that their definition of the number 1 runs to some tens of thousands of symbols. We show that that is a considerable under-estimate, the true number of symbols being that in the title, not counting 1 179 618 517 981 links between symbols that are needed to disambiguate the whole expression.}}</ref>
 | source = Michael Barany<ref>{{cite journal |last1=Barany |first1=Michael |title=The Mathematical Pranksters behind Nicolas Bourbaki |journal=JSTOR Daily |date=24 March 2021 |url=https://daily.jstor.org/the-mathematical-pranksters-behind-nicolas-bourbaki/ |access-date=5 January 2024}}</ref>
 | width  = 30%
 | align  = right
}}

The content of the ''Éléments'' is divided into ''books''—major topics of discussion, ''volumes''—individual, physical books, and ''chapters'', together with certain summaries of results, historical notes, and other details. The volumes of the ''Éléments'' have had a complex publication history. Material has been revised for new editions, published chronologically out of order of its intended logical sequence, grouped together and partitioned differently in later volumes, and translated into English. For example, the second book on ''Algebra'' was originally released in eight French volumes: the first in 1942 being chapter 1 alone, and the last in 1980 being chapter 10 alone. This presentation was later condensed into five volumes with chapters 1–3 in the first volume, chapters 4–7 in the second, and chapters 8–10 each remaining the third through fifth volumes of that portion of the work.<ref name="archiver" /> The English edition of Bourbaki's ''Algebra'' consists of translations of the three volumes consisting of chapters 1–3, 4–7 and 8, with chapters 9 and 10 unavailable in English as of {{year}}.

When Bourbaki's founders began working on the ''Éléments'', they originally conceived of it as a "treatise on analysis", the proposed work having a working title of the same name (''Traité d'analyse''). The opening part was to comprehensively deal with the [[foundations of mathematics]] prior to analysis, and was referred to as the "Abstract Packet". Over time, the members developed this proposed "opening section" of the work to the point that it would instead run for several volumes and comprise a major part of the work, covering set theory, abstract algebra, and topology. Once the project's scope expanded far beyond its original purpose, the working title ''Traité d'analyse'' was dropped in favor of ''Éléments de mathématique''.{{sfn|Mashaal|p=11}} The unusual, singular "Mathematic" was meant to connote Bourbaki's belief in the unity of mathematics.{{sfn|Aczel|pp=99–100}}{{sfn|Borel|p=374}}{{sfn|Mashaal|p=55}}  The first six books of the ''Éléments'', representing the first half of the work, are numbered sequentially and ordered logically, with a given statement being established only on the basis of earlier results.<ref>''Theory of Sets'', pp. v-vi.</ref>  This first half of the work bore the subtitle ''Les structures fondamentales de l’analyse'' (''Fundamental Structures of Analysis''),<ref name="archiver" />{{sfn|Mashaal|p=83}}<ref name="Boyer">{{cite book |author-link1=Carl Benjamin Boyer |last1=Boyer |first1=Carl B. |author-link2=Uta Merzbach |last2=Merzbach |first2=Uta C. |contribution=foreword |contributor-link=Isaac Asimov |contributor-last=Asimov |contributor-first=Isaac |title=A History of Mathematics |date=20 March 1991 |edition=Second |publisher=Wiley |isbn=9780471543978 |page=629}}</ref> covering established mathematics (algebra, analysis) in the group's style.  The second half of the work consists of unnumbered books treating modern areas of research (Lie groups, commutative algebra), each presupposing the first half as a shared foundation but without dependence on each other.  This second half of the work, consisting of newer research topics, does not have a corresponding subtitle.

The volumes of the ''Éléments'' published by Hermann were indexed by chronology of publication and referred to as ''fascicules'': installments in a large work. Some volumes did not consist of the normal definitions, proofs, and exercises in a math textbook, but contained only summaries of results for a given topic, stated without proof. These volumes were referred to as ''Fascicules de résultats'', with the result that ''fascicule'' may refer to a volume of Hermann's edition, or to one of the "summary" sections of the work (e.g. ''Fascicules de résultats'' is translated as "Summary of Results" rather than "Installment of Results", referring to the content rather than a specific volume).{{efn|The [[History of mathematics|mathematical historian]] [[Leo Corry]] also observed that the phrase "Summary of Results" is a misleading one for a distinct reason, instead referring to the content of the ''Éléments'' rather than the publication history of its volumes.{{sfn|Corry|1992|p=326}}{{sfn|Corry|2004|p=320}} }} The first volume of Bourbaki's ''Éléments'' to be published was the Summary of Results in the ''Theory of Sets'', in 1939.{{sfn|Senechal|pp=22–28}}<ref name="archiver" />{{sfn|Mashaal|p=52}} Similarly one of the work's later books, ''Differential and Analytic Manifolds'', consisted only of two volumes of summaries of results, with no chapters of content having been published.

Later installments of the ''Éléments'' appeared infrequently during the 1980s and 1990s. A volume of ''Commutative Algebra'' (chapters 8–9) was published in 1983, and no other volumes were issued until the appearance of the same book's tenth chapter in 1998. During the 2010s, Bourbaki increased its productivity. A re-written and expanded version of the eighth chapter of ''Algebra'' appeared in 2012, the first four chapters of a new book treating [[Algebraic Topology]] was published in 2016, and the first two chapters of a revised and expanded edition of ''Spectral Theory'' was issued in 2019 while the remaining three (completely new) chapters appeared in 2023.

[[File:Bourbaki, Theorie des ensembles maitrier.jpg|thumb|First book of the ''[[Éléments de mathématique]]'', 1970 edition]]

{| class="wikitable sortable" style="text-align: center"
|+''Éléments de mathématique''<ref name="archiver" />{{efn|Years refer to the date of publication of each book's first volume, which also contains its first proper chapter.  There are two exceptions: the first published installment of the ''Theory of Sets'' was a summary of results in 1939, and its first proper chapter did not appear until 1954.  For ''Differential and Analytic Manifolds'', only a two-volume summary of results was published in 1967 and 1971, with no proper chapters appearing.}}
!Year
!Book
!References
|-
|1954
|''Theory of Sets''
|<ref>{{cite journal |last=Bagemihl |first=Frederick |author-link=Frederick Bagemihl |title=Review: ''Théorie des ensembles'' (Chapter III) |journal=[[Bulletin of the American Mathematical Society]] |year=1958 |volume=64 |issue=6 |pages=390–91 |url=https://www.ams.org/journals/bull/1958-64-06/S0002-9904-1958-10248-7/S0002-9904-1958-10248-7.pdf |doi=10.1090/s0002-9904-1958-10248-7 |doi-access=free }}</ref>
|-
|1942
|''Algebra''
|<ref name="Artin">{{cite journal |last=Artin |first=Emil |author-link=Emil Artin |title=Review: ''Éléments de mathématique'', by N. Bourbaki, Book II, ''Algebra''. Chaps. I–VII |journal=Bulletin of the American Mathematical Society |year=1953 |volume=59 |issue=5 |pages=474–79 |url=https://www.ams.org/journals/bull/1953-59-05/S0002-9904-1953-09725-7/S0002-9904-1953-09725-7.pdf |doi=10.1090/s0002-9904-1953-09725-7 |doi-access=free }}</ref><ref>{{cite journal |last=Rosenberg |first= Alex |author-link=Alex F. T. W. Rosenberg |title=Review: ''Éléments de mathématiques'' by N. Bourbaki. Book II, Algèbre. Chapter VIII, ''Modules et anneaux semi-simples'' |journal=Bulletin of the American Mathematical Society |year=1960 |volume=66 |issue=1 |pages=16–19 |url=https://www.ams.org/journals/bull/1960-66-01/S0002-9904-1960-10371-0/S0002-9904-1960-10371-0.pdf |doi=10.1090/S0002-9904-1960-10371-0 |doi-access=free }}</ref><ref>{{cite journal |last=Kaplansky |first=Irving |author-link=Irving Kaplansky |title=Review: ''Formes sesquilinéairies et formes quadratiques'' by N. Bourbaki, ''Éléments de mathématique'' I, Livre II |journal=Bulletin of the American Mathematical Society |year=1960 |volume=66 |issue=4 |pages=266–67 |url=https://www.ams.org/journals/bull/1960-66-04/S0002-9904-1960-10461-2/S0002-9904-1960-10461-2.pdf|doi=10.1090/s0002-9904-1960-10461-2 |doi-access=free }}</ref>
|-
|1940
|''General Topology''
|
|-
|1949
|''Functions of a Real Variable''
|
|-
|1953
|''Topological Vector Spaces''
|
|-
|1952
|''Integration''
|<ref>{{cite journal |last=Halmos |first=Paul |author-link=Paul Halmos |title=Review: ''Intégration'' (Chap. I–IV) by N. Bourbaki |journal=Bulletin of the American Mathematical Society |year=1953 |volume=59 |issue=3 |pages=249–55 |url=https://www.ams.org/journals/bull/1953-59-03/S0002-9904-1953-09698-7/S0002-9904-1953-09698-7.pdf |doi=10.1090/S0002-9904-1953-09698-7 |doi-access=free }}</ref><ref>{{cite journal |last=Munroe |first=M. E. |title=Review: ''Intégration'' (Chapter V) by N. Bourbaki |journal=Bulletin of the American Mathematical Society |year=1958 |volume=64 |issue=3 |pages=105–06 |url=https://www.ams.org/journals/bull/1958-64-03/S0002-9904-1958-10176-7/S0002-9904-1958-10176-7.pdf |doi=10.1090/s0002-9904-1958-10176-7 |doi-access=free }}</ref>
|-
|1960
|''Lie Groups and Lie Algebras''
|
|-
|1961
|''Commutative Algebra''
|<ref>{{cite journal |last=Nagata |first=Masayoshi |author-link=Masayoshi Nagata |title=''Éléments de mathématique. Algèbre commutative'', by N. Bourbaki, Chapitres 8 et 9 |journal=Bulletin of the American Mathematical Society |series=New Series |year=1985 |volume=12 |issue=1 |pages=175–77 |url=https://www.ams.org/journals/bull/1985-12-01/S0273-0979-1985-15338-8/S0273-0979-1985-15338-8.pdf |doi=10.1090/s0273-0979-1985-15338-8 |doi-access=free }}</ref>
|-
|1967
|''Spectral Theory''
|
|-
|1967
|''Differential and Analytic Manifolds''
|
|-
|2016
|''Algebraic Topology''
|<ref>{{Cite book |url=https://www.springer.com/us/book/9783662493601|title=Topologie Algébrique, Chapitres 1 à 4 |last=Bourbaki |first=Nicolas |year=2016 |publisher=Springer |doi=10.1007/978-3-662-49361-8 |isbn=978-3-662-49360-1 |access-date=2016-02-08 }}</ref>
|-
|1960
|''Elements of the History of Mathematics''
|
|}

===Séminaire Bourbaki===
{{main|Séminaire Bourbaki}}

The Séminaire Bourbaki has been held regularly since 1948, and lectures are presented by non-members and members of the collective. As of {{year}} the Séminaire Bourbaki has run to over a thousand recorded lectures in its written incarnation, denoted chronologically by simple numbers.<ref name="seminar">{{cite web |url=http://www.bourbaki.ens.fr/Editeurs.html |title=Éditeurs du Séminaire |publisher=Association des collaborateurs de Nicolas Bourbaki}}</ref> At the time of a June 1999 lecture given by Jean-Pierre Serre on the topic of Lie groups, the total lectures given in the series numbered 864, corresponding to roughly 10,000 pages of printed material.{{sfn|Mashaal|pp=98–99}}

===Articles===

[[File:Kosambi-dd.jpg|thumb|right|150px|[[Damodar Dharmananda Kosambi|Damodar Kosambi]] authored the first article attributing material to "Bourbaki"]]

Several journal articles have appeared in the mathematical literature with material or authorship attributed to Bourbaki; unlike the ''Éléments'', they were typically written by individual members<ref name="archiver" /> and not crafted through the usual process of group consensus. Despite this, Jean Dieudonné's essay "The Architecture of Mathematics" has become known as Bourbaki's [[manifesto]].{{sfn|Aubin|pp=305–08}}{{sfn|Corry|1997|pp=272–73}} Dieudonné addressed the issue of overspecialization in mathematics, to which he opposed the inherent unity of ''mathematic'' (as opposed to mathematics) and proposed mathematical structures as useful tools which can be applied to several subjects, showing their common features.{{sfn|Corry|2004|pp=303–05}} To illustrate the idea, Dieudonné described three different systems in arithmetic and geometry and showed that all could be described as examples of a [[group (mathematics)|group]], a specific kind of ([[algebraic structure|algebraic]]) structure.{{sfn|Bourbaki 1950|pp=224–26}} Dieudonné described the [[axiomatic method]] as "the '[[Taylor system]]' for mathematics" in the sense that it could be used to solve problems efficiently.{{sfn|Bourbaki 1950|p=227}}{{efn|Dieudonné immediately qualified the comparison as "a very poor analogy", continuing: "the mathematician does not work like a machine, nor as the workingman on a moving belt; we can not over-emphasize the fundamental role played in his research by a special intuition, which is not the popular sense-intuition, but rather a kind of direct divination... of the normal behavior... of mathematical beings."{{sfn|Bourbaki 1950|p=227}} }} Such a procedure would entail identifying relevant structures and applying established knowledge about the given structure to the specific problem at hand.{{sfn|Bourbaki 1950|p=227}}
* {{cite journal |last=Kosambi |first=D.D. |title=On a Generalization of the Second Theorem of Bourbaki |journal=Bulletin of the Academy of Sciences of the United Provinces of Agra and Oudh, Allahabad, India |volume=1 |pages=145–47 |date=1931 |isbn=978-81-322-3674-0 }} Reprinted in {{cite book|title= D.D. Kosambi: Selected Works in Mathematics and Statistics|publisher=Springer|year=2016|doi=10.1007/978-81-322-3676-4_6|editor-first=Ramakrishna|editor-last=Ramaswamy|pages=55–57}} Kosambi attributed material in the article to "D. Bourbaki", the first mention of the eponymous Bourbaki in the literature.
* {{cite journal |url=https://gallica.bnf.fr/ark:/12148/bpt6k31534/f1311.image |title=Sur un théorème de Carathéodory et la mesure dans les espaces topologiques |last=Bourbaki |first=Nicolas |journal=[[Comptes rendus de l'Académie des Sciences]] |volume=201 |pages=1309–11 |date=1935}} Presumptive author: André Weil.
* {{cite journal |author-mask=2 |url=https://gallica.bnf.fr/ark:/12148/bpt6k3158p/f1701.image |title=Sur les espaces de Banach |last=Bourbaki |first=Nicolas |journal=Comptes rendus de l'Académie des Sciences |volume=206 |pages=1701–04 |date=1938}} Presumptive author: Jean Dieudonné.
* {{cite journal |author-mask=2 |title=Note de tératopologie II |last1=Bourbaki |first1=Nicolas |last2=Dieudonné |first2=Jean |journal=Revue scientifique (Or, "Revue rose") |pages=180–81 |date=1939}} Presumptive author: Jean Dieudonné. Second in a series of three articles.
* {{cite journal |author-mask=2 |url=https://gallica.bnf.fr/ark:/12148/bpt6k3164q/f215.image |title=Espaces minimaux et espaces complètement séparés |last=Bourbaki |first=Nicolas |journal=Comptes rendus de l'Académie des Sciences |volume=212 |pages=215–18 |date=1941}} Presumptive author: Jean Dieudonné or André Weil.
* {{cite book    |author-mask=2 |title=Les grands courants de la pensée mathématique |last=Bourbaki |first=Nicolas |chapter=L'architecture des mathématiques |editor-last=[[François Le Lionnais|Le Lionnais]] |editor-first=François |publisher=Actes Sud |pages=35–47 |date=1948}} Presumptive author: Jean Dieudonné.
* {{cite journal |author-mask=2 |title=Foundations of Mathematics for the Working Mathematician |last=Bourbaki |first=Nicolas |journal=[[Journal of Symbolic Logic]] |volume=14 |issue=1 |pages=1–8 |date=1949|jstor=2268971 |doi=10.2307/2268971 |s2cid=26516355 }} Presumptive author: André Weil.
* {{cite journal |author-mask=2 |title=Sur le théorème de Zorn |last=Bourbaki |first=Nicolas |s2cid=117826806 |journal=[[Archiv der Mathematik]] |volume=2 |issue=6 |pages=433–37 |date=1949|doi=10.1007/BF02036949 }} Presumptive author: Henri Cartan or Jean Dieudonné.
* {{cite journal |author-mask=2 |title=The Architecture of Mathematics |last=Bourbaki |first=Nicolas |journal=[[American Mathematical Monthly]] |volume=57 |issue=4 |pages=221–32 |date=1950|jstor=2305937 |doi=10.1080/00029890.1950.11999523 }} Presumptive author: Jean Dieudonné. Authorized translation of the book chapter ''L'architecture des mathématiques'', appearing in English as a journal article.
* {{cite journal |author-mask=2 |url=http://www.numdam.org/item/?id=AIF_1950__2__5_0 |title=Sur certains espaces vectoriels topologiques |last=Bourbaki |first=Nicolas |journal=[[Annales de l'Institut Fourier]] |volume=2 |pages=5–16 |date=1950 |doi = 10.5802/aif.16 |doi-access=free }} Presumptive authors: Jean Dieudonné and Laurent Schwartz.

===''La Tribu''===

''La Tribu'' is Bourbaki's internal newsletter, distributed to current and former members. The newsletter usually documents recent conferences and activity in a humorous, informal way, sometimes including poetry.{{sfn|Mashaal|pp=108–11}} Member [[Pierre Samuel]] wrote the newsletter's narrative sections for several years.{{sfn|Beaulieu|1999|p=234}} Early editions of ''La Tribu'' and related documents have been made publicly available by Bourbaki.<ref name="barchive" />

Historian Liliane Beaulieu examined ''La Tribu'' and Bourbaki's other writings, describing the group's humor and private language as an "art of memory" which is specific to the group and its chosen methods of operation.{{sfn|Beaulieu|1999|p=224}} Because of the group's secrecy and informal organization, individual memories are sometimes recorded in a fragmentary way, and may not have significance to other members.{{sfn|Beaulieu|1999|pp=231–32}} On the other hand, the predominantly French, ENS background of the members, together with stories of the group's early period and successes, create a shared culture and mythology which is drawn upon for group identity. ''La Tribu'' usually lists the members present at a conference, together with any visitors, family members or other friends in attendance. Humorous descriptions of location or local "props" (cars, bicycles, binoculars, etc.) can also serve as [[mnemonic]] devices.{{sfn|Beaulieu|1999|p=226}}

==Membership==

As of 2000, Bourbaki has had "about forty" members.{{sfn|Mashaal|p=18}} Historically the group has numbered about ten{{sfn|Beaulieu|1999|p=220}} to twelve{{sfn|Senechal|pp=22–28}} members at any given point, although it was briefly (and officially) limited to nine members at the time of founding.{{sfn|Mashaal|p=8}} Bourbaki's membership has been described in terms of generations:

{{Blockquote|text=Bourbaki was always a very small group of mathematicians, typically numbering about twelve people.  Its first generation was that of the founding fathers, those who created the group in 1934: Weil, Cartan, Chevalley, Delsarte, de Possel, and Dieudonné.  Others joined the group, and others left its ranks, so that some years later there were about twelve members, and that number remained roughly constant.  Laurent Schwartz was the only mathematician to join Bourbaki during the war, so his is considered an intermediate generation.  After the war, a number of members joined: [[Jean-Pierre Serre]], [[Pierre Samuel]], [[Jean-Louis Koszul]], [[Jacques Dixmier]], [[Roger Godement]], and [[Samuel Eilenberg|Sammy Eilenberg]].  These people constituted the second generation of Bourbaki.  In the 1950s, the third generation of mathematicians joined Bourbaki.  These people included [[Alexandre Grothendieck]], [[François Bruhat]], [[Serge Lang]], the American mathematician [[John Tate (mathematician)|John Tate]], [[Pierre Cartier (mathematician)|Pierre Cartier]], and the Swiss mathematician [[Armand Borel]].{{sfn|Senechal|pp=22–28}}{{sfn|Aczel|pp=108–09}} }}

After the first three generations there were roughly twenty later members, not including current participants. Bourbaki has a custom of keeping its current membership secret, a practice meant to ensure that its output is presented as a collective, unified effort under the Bourbaki pseudonym, not attributable to any one author (e.g. for purposes of copyright or royalty payment). This secrecy is also intended to deter unwanted attention which could disrupt normal operations. However, former members freely discuss Bourbaki's internal practices upon departure.{{sfn|Senechal|pp=22–28}}{{sfn|Mashaal|p=14}}

Prospective members are invited to conferences and styled as [[guinea pig]]s, a process meant to vet the newcomer's mathematical ability.{{sfn|Senechal|pp=22–28}}{{sfn|Mashaal|p=16}} In the event of agreement between the group and the prospect, the prospect eventually becomes a full member.{{efn|Examples of guinea pigs who attended conferences without necessarily joining include one "Mirlès", who attended the official founding conference in Besse-en-Chandesse, [[Marcel Berger]], [[Jean Giraud (mathematician)|Jean Giraud]], [[Bernard Malgrange]], and [[René Thom]].{{sfn|Aubin|p=330}}{{sfn|Beaulieu|1999|p=242}}{{sfn|Mashaal|pp=9, 109, 130}}  Other guinea pigs and visitors have also been listed.<ref>{{cite web |url=http://archives-bourbaki.ahp-numerique.fr/references/item-type-metadata-present |title=Membres présents aux réunions |website=Archives Bourbaki }}</ref> }} The group is supposed to have an age limit: active members are expected to retire at (or about) 50 years of age.{{sfn|Senechal|pp=22–28}}{{sfn|Guedj|p=18}} At a 1956 conference, Cartan read a letter from Weil which proposed a "gradual disappearance" of the founding members, forcing younger members to assume full responsibility for Bourbaki's operations.<ref name="numericana" />{{sfn|Mashaal|pp=18–19}} This rule is supposed to have resulted in a complete change of personnel by 1958.<ref name="Mainard"/> However, historian Liliane Beaulieu has been critical of the claim. She reported never having found written affirmation of the rule,{{sfn|Aubin|p=298}} and has indicated that there have been exceptions.{{sfn|Beaulieu|1999|p=248}} The age limit is thought to express the founders' intent that the project should continue indefinitely, operated by people at their best mathematical ability—in the mathematical community, there is a widespread belief that mathematicians produce their best work while young.{{sfn|Mashaal|pp=18–19}}{{sfn|Aubin|p=304}} Among full members there is no official hierarchy; all operate as equals, having the ability to interrupt conference proceedings at any point, or to challenge any material presented. However, André Weil has been described as "first among equals" during the founding period, and was given some deference.{{sfn|Mashaal|p=12}} On the other hand, the group has also poked fun at the idea that older members should be afforded greater respect.{{sfn|Mashaal|pp=111–12}}

Bourbaki conferences have also been attended by members' family, friends, visiting mathematicians, and other non-members of the group.{{efn|In 1948 one Nicolaidis Bourbaki, a diplomat and relative of the eponymous French general, sought the group out to understand why the family name had been taken.  The diplomat and the mathematical collective met on friendly terms, and Nicolaidis was a dinner guest at some of the group's conferences.{{sfn|Beaulieu|1999|p=236}}{{sfn|Mashaal|pp=29, 33}} }} Bourbaki is not known ever to have had any female members.{{sfn|Guedj|p=18}}{{sfn|Beaulieu|1999|p=220}}

[[File:Jean Dieudonné.jpg|thumb|right|130px|[[Jean Dieudonné]], founding member]]

[[File:Jean-Pierre Serre (2009) cropped.jpg|thumb|right|130px|[[Jean-Pierre Serre]], second generation member]]

[[File:Alexander Grothendieck.jpg|thumb|right|130px|[[Alexander Grothendieck]], third generation member, left Bourbaki largely over a disagreements about incorporation of [[category theory]] in the treatise{{sfn|Aczel|p=205}}{{sfn|Krömer|2006|pp=149-150}}{{sfn|Mashaal|pp=81-84}}{{sfn|Corry|2009|pp=581-584}}]]

[[File:Armand Borel 1975 (headshot).jpg|thumb|right|130px|[[Armand Borel]], third generation member]]

[[File:Hyman Bass 1978 (portioned B).jpg|thumb|right|130px|[[Hyman Bass]], later member]]

{|class="wikitable sortable" style="text-align: center"
|+Former members of the Nicolas Bourbaki collective<ref name="numericana" />{{sfn|Aczel|pp=87, 108–09}}{{sfn|Mashaal|pp=6, 8, 12, 17–18, 60, 100, 105}}
!data-sort-type=number colspan="2"| Generation
!data-sort-type=text|Name
!data-sort-type=number|Born
!data-sort-type=number|ENS{{efn|Dates refer to [[Matriculation|entrance into the university]], not graduation.}}
!data-sort-type=number|Joined{{efn|Bourbaki's secrecy and informality have made it difficult to establish members' dates of joining and leaving.  For past members with uncertain dates, it has been suggested that the members' periods of [[Floruit|flourishing]] ({{circa|age 25–50}}) is the best available estimate.{{sfn|Mashaal|pp=18–19}} }}{{efn|Some members attended conferences as guinea pigs for a period of years before becoming full members.  Armand Borel began attending Bourbaki conferences c. 1949, becoming a full member c. 1953 and departing in 1973.{{sfn|Borel|pp=373–75}}  Pierre Cartier first attended a Bourbaki conference as a guinea pig in 1951, becoming a full member in 1955 and departing in 1983.{{sfn|Senechal|pp=22–28}}{{sfn|Aczel|pp=105–08}}  Where sources make a distinction, the date of full membership is given or approximated. }}
!data-sort-type=number|Left
!data-sort-type=number|Died
|-
|data-sort-value=1 rowspan="11"|First{{efn|The collective's founding generation included a core group of five{{sfn|Borel|p=374}} who led its activities and established its norms, remaining active for several years.  Another six minor members participated on shorter-term bases during its earliest days, ranging from a few months to a few years.}}
|rowspan="5"|Core members
|data-sort-value="Cartan"|[[Henri Cartan]]
|1904
|1923
|1934
|data-sort-value=1958|c. 1956–58
|2008
|-
|data-sort-value="Chevalley"|[[Claude Chevalley]]
|1909
|1926
|1934
|data-sort-value=1958|c. 1956–58
|1984
|-
|data-sort-value="Delsarte"|[[Jean Delsarte]]
|1903
|1922
|1934
|data-sort-value=1958|c. 1956–58
|1968
|-
|data-sort-value="Dieudonné"|[[Jean Dieudonné]]
|1906
|1924
|1934
|data-sort-value=1958|c. 1956–58
|1992
|-
|data-sort-value="Weil"|[[André Weil]]
|1906
|1922
|1934
|data-sort-value=1958|c. 1956–58
|1998
|-
|rowspan="6"|Minor members
|data-sort-value="Coulomb"|[[Jean Coulomb]]
|1904
|1923
|1935
|1937
|1999
|-
|data-sort-value="Dubreil"|[[Paul Dubreil]]
|1904
|1923
|1935
|1935
|1994
|-
|data-sort-value="Ehresmann"|[[Charles Ehresmann]]
|1905
|1924
|1935
|1950
|1979
|-
|data-sort-value="Leray"|[[Jean Leray]]
|1906
|1926
|1935
|1935
|1998
|-
|data-sort-value="Mandelbrojt"|[[Szolem Mandelbrojt]]
|1899
|—
|1935
|—
|1983
|-
|data-sort-value="Possel"|[[René de Possel]]
|1905
|1923
|1934
|—
|1974
|-
|data-sort-value=2 rowspan="7" colspan="2"|Second{{efn|Aczel described Schwartz as an inter-generational member, the only one to join during the Second World War.  However Schwartz did not participate in the group's founding.}}
|data-sort-value="Dixmier"|[[Jacques Dixmier]]
|1924
|1942
|—
|—
|—
|-
|data-sort-value="Eilenberg"|[[Samuel Eilenberg]]
|1913
|—
|data-sort-value="1951"|c. 1951
|1966
|1998
|-
|data-sort-value="Godement"|[[Roger Godement]]
|1921
|1940
|—
|—
|2016
|-
|data-sort-value="Koszul"|[[Jean-Louis Koszul]]
|1921
|1940
|—
|—
|2018
|-
|data-sort-value="Samuel"|[[Pierre Samuel]]
|1921
|1940
|1947
|1971
|2009
|-
|data-sort-value="Schwartz"|[[Laurent Schwartz]]
|1916
|1934
|—
|—
|2002
|-
|data-sort-value="Serre"|[[Jean-Pierre Serre]]
|1926
|1945
|—
|—
|—
|-
|data-sort-value=3 rowspan="6" colspan="2"|Third
|data-sort-value="Borel"|[[Armand Borel]]
|1923
|—
|data-sort-value=1953|c. 1953
|1973
|2003
|-
|data-sort-value="Bruhat"|[[François Bruhat]]
|1929
|1948
|—
|—
|2007
|-
|data-sort-value="Cartier"|[[Pierre Cartier (mathematician)|Pierre Cartier]]
|1932
|1950
|1955
|1983
|2024
|-
|data-sort-value="Grothendieck"|[[Alexander Grothendieck]]
|1928
|1945
|1955
|1960
|2014
|-
|data-sort-value="Lang"|[[Serge Lang]]
|1927
|—
|—
|—
|2005
|-
|data-sort-value="Tate"|[[John Tate (mathematician)|John Tate]]
|1925
|—
|—
|—
|2019
|-
|data-sort-value=4 rowspan="24" colspan="2"|Later members{{efn|Most other members were born after the above three generations and were therefore active in the group at later dates.  However, two were born contemporaries of the founding generation: Charles Pisot in 1909, and Claude Chabauty in 1910.}}{{efn|Cartier and Aczel also described a fourth generation of Bourbaki members (as opposed to later members in general), former students of Grothendieck who joined during the 1960s.{{sfn|Senechal|pp=22–28}}{{sfn|Aczel|p=119}}  This may refer to those of Grothendieck's doctoral students who later became Bourbaki members, such as [[Michel Demazure]] and [[Jean-Louis Verdier]].<ref name="genealogy">{{cite web |url=https://genealogy.math.ndsu.nodak.edu/id.php?id=31245 |title=Alexander Grothendieck |website=Mathematics Genealogy Project}}</ref>}}
|data-sort-value="Bass"|[[Hyman Bass]]
|1932
|—
|—
|—
|—
|-
|data-sort-value="Beauville"|[[Arnaud Beauville]]
|1947
|1966
|—
|1997
|—
|-
|data-sort-value="Ben Arous"|[[Gérard Ben Arous]]
|1957
|1977
|—
|—
|—
|-
|data-sort-value="Bennequin"|[[Daniel Bennequin]]
|1952
|1972
|—
|—
|—
|-
|data-sort-value="Chabauty"|[[Claude Chabauty]]
|1910
|1929
|—
|—
|1990
|-
|data-sort-value="Connes"|[[Alain Connes]]
|1947
|1966
|—
|—
|—
|-
|data-sort-value="Demazure"|[[Michel Demazure]]
|1937
|1955
|—
|data-sort-value="1985"|c. 1985
|—
|-
|data-sort-value="Douady"|[[Adrien Douady]]
|1935
|1954
|—
|—
|2006
|-
|data-sort-value="Gérard"|[[:fr:Patrick Gérard (mathématicien)|Patrick Gérard]]<small>[[:fr:Wikipédia:Accueil_principal|[fr]]]</small>
|1961
|1981
|—
|—
|—
|-
|data-sort-value="Henniart"|[[Guy Henniart]]
|1953
|1973
|—
|—
|—
|-
|data-sort-value="Illusie"|[[Luc Illusie]]
|1940
|1959
|—
|—
|—
|-
|data-sort-value="Julg"|Pierre Julg
|1959
|1977
|—
|—
|—
|-
|data-sort-value="Lebeau"|[[Gilles Lebeau]]
|1954
|1974
|—
|—
|—
|-
|data-sort-value="Martineau"|[[André Martineau]]
|1930
|1949
|—
|—
|1972
|-
|data-sort-value="Mathieu"|Olivier Mathieu
|1960
|1980
|1989
|c. 2010
|—
|-
|data-sort-value="Monvel"|[[Louis Boutet de Monvel]]
|1941
|1960
|1971
|1991
|2014
|-
|data-sort-value="Oesterlé"|[[Joseph Oesterlé]]
|1954
|1973
|—
|—
|—
|-
|data-sort-value="Pisot"|[[Charles Pisot]]
|1909
|1929
|—
|—
|1984
|-
|data-sort-value="Raynaud"|[[Michel Raynaud]]
|1938
|1958
|—
|—
|2018
|-
|data-sort-value="Rosso"|Marc Rosso
|1962
|1982
|—
|—
|—
|-
|data-sort-value="Skandalis"|[[Georges Skandalis]]
|1955
|1975
|—
|—
|—
|-
|data-sort-value="Teissier"|[[Bernard Teissier]]
|1945
|—
|—
|—
|—
|-
|data-sort-value="Verdier"|[[Jean-Louis Verdier]]
|1937
|1955
|—
|—
|1989
|-
|data-sort-value="Yoccoz"|[[Jean-Christophe Yoccoz]]
|1957
|1975
|data-sort-value="1995"|c. 1995
|data-sort-value="1995"|c. 1995
|2016
|}

==Influence and criticism==

{{see also|Structuralism|Structuralism (philosophy of mathematics)}}

Bourbaki was influential in 20th century mathematics and had some interdisciplinary impact on the humanities and the arts, although the extent of the latter influence is a matter of dispute. The group has been praised and criticized for its method of presentation, its working style, and its choice of mathematical topics.

===Influence===

{{multiple image
|align=right
|direction=vertical
|width=150
|image1=Empty set symbol.svg
|image2=Borne Michelin Virages.JPG
|caption2=Bourbaki introduced notations for the [[empty set]], as well as a [[Bourbaki dangerous bend symbol|dangerous bend symbol]] meant to indicate difficult material
}}

Bourbaki introduced several mathematical notations which have remained in use. Weil took the letter {{large|[[Ø]]}} of the Norwegian alphabet and used it to denote the [[empty set]], {{large|<math>\varnothing</math>}}.<ref>[http://jeff560.tripod.com/set.html Earliest Uses of Symbols of Set Theory and Logic.]</ref> This notation first appeared in the Summary of Results on the ''Theory of Sets'',<ref name="set theory">{{cite book |last=Bourbaki |first=Nicolas |title=Theory of Sets |publisher=Springer |isbn=9783540225256 |pages=72, 349 |year=2004}}</ref> and remains in use. The words [[injective]], [[surjective]] and [[Bijection|bijective]] were introduced to refer to [[Function (mathematics)|functions]] which satisfy certain properties.<ref>''Theory of Sets'', p. 84.</ref><ref name="Gunderman">{{Cite web|url=http://theconversation.com/nicolas-bourbaki-the-greatest-mathematician-who-never-was-122845|title=Nicolas Bourbaki: The greatest mathematician who never was|last=Gunderman|first=David|website=The Conversation|date=7 December 2019 |language=en|access-date=2019-12-14}}</ref> Bourbaki used simple language for certain geometric objects, naming them ''pavés'' ([[paving stone]]s) and ''boules'' ([[Ball (mathematics)|balls]]) as opposed to "[[Parallelohedron|parallelotopes]]" or "[[Hypersphere|hyperspheroids]]".{{sfn|Mashaal|p=105}} Similarly in its treatment of topological vector spaces, Bourbaki defined a [[Barrelled space|barrel]] as a set which is [[convex set|convex]], [[balanced set|balanced]], [[absorbing set|absorbing]], and [[closed set|closed]].<ref name="tvs">{{cite book |last=Bourbaki |first=Nicolas |translator-last1=Eggleston |translator-first1=H.G. |translator-last2=Madan |translator-first2=S. |title=Topological Vector Spaces: Chapters 1-5 |year=1987 |publisher=Springer |isbn=9783540423386}} Chapter III, p. 24.</ref> The group were proud of this definition, believing that the shape of a [[wine barrel]] typified the mathematical object's properties.{{sfn|Beaulieu|1999|p=228}}{{sfn|Mashaal|pp=107–08}} Bourbaki also employed a "[[Bourbaki dangerous bend symbol|dangerous bend]]" symbol {{large|☡}} in the margins of its text to indicate an especially difficult piece of material. Bourbaki enjoyed its greatest influence during the 1950s and 1960s, when installments of the ''Éléments'' were published frequently.

Bourbaki had some interdisciplinary influence on other fields, including [[anthropology]] and [[psychology]]. This influence was in the context of [[structuralism]], a school of thought in the [[humanities]] which stresses the relationships between objects over the objects themselves, pursued in various fields by other French intellectuals. In 1943, André Weil met the anthropologist [[Claude Lévi-Strauss]] in New York, where the two undertook a brief collaboration. At Lévi-Strauss' request, Weil wrote a brief appendix describing marriage rules for four classes of people within [[Aboriginal Australian]] society, using a [[mathematical model]] based on [[group theory]].{{sfn|Aczel|pp=129–48}}{{sfn|Aubin|pp=308–11}} The result was published as an appendix in Lévi-Strauss' [[Claude Lévi-Strauss#Expatriation|''Elementary Structures of Kinship'']], a work examining family structures and the [[incest taboo]] in human cultures.<ref name="kinship">{{cite book |title=The Elementary Structures of Kinship |via=[[Internet Archive]] |last=Weil |first=André |chapter=Chapter XIV: Appendix to Part One |editor-last=Lévi-Strauss |editor-first=Claude |chapter-url=https://archive.org/details/TheElementaryStructuresOfKinshipLeviStrauss |year=1971 |pages=[https://archive.org/details/TheElementaryStructuresOfKinshipLeviStrauss/page/n255 221–29]}}</ref> In 1952, Jean Dieudonné and [[Jean Piaget]] participated in an interdisciplinary conference on mathematical and mental structures. Dieudonné described mathematical "mother structures" in terms of Bourbaki's project: composition, neighborhood, and order.{{sfn|Aczel|pp=161–64}} Piaget then gave a talk on children's mental processes, and considered that the psychological concepts he had just described were very similar to the mathematical ones just described by Dieudonné.{{sfn|Aczel|p=162}}{{sfn|Mashaal|p=73}} According to Piaget, the two were "impressed with each other".{{sfn|Aubin|p=318}} The psychoanalyst [[Jacques Lacan]] liked Bourbaki's collaborative working style and proposed a similar collective group in psychology, an idea which did not materialize.{{sfn|Aczel|p=169}}

Bourbaki was also cited by [[Post-structuralism|post-structuralist]] philosophers. In their joint work ''[[Anti-Oedipus]]'', [[Gilles Deleuze]] and [[Félix Guattari]] presented a [[criticism of capitalism]]. The authors cited Bourbaki's use of the axiomatic method (with the purpose of establishing truth) as a distinct counter-example to [[management]] processes which instead seek [[economic efficiency]]. The authors said of Bourbaki's axiomatics that "they do not form a Taylor system", inverting the phrase used by Dieudonné in "The Architecture of Mathematics".{{sfn|Bourbaki 1950|p=227}}<ref name="oedipus">{{cite book |last1=Deleuze |first1=Gilles |last2=Guattari |first2=Félix |title=Anti-Oedipus |url=https://archive.org/details/antioedipuscapit00dele_367 |url-access=limited |date=1972 |publisher=[[University of Minnesota Press]] |isbn=978-0816612253 |page=[https://archive.org/details/antioedipuscapit00dele_367/page/n270 251] }}</ref> In ''[[The Postmodern Condition]]'', [[Jean-François Lyotard]] criticized the "legitimation of knowledge", the process by which statements become accepted as valid. As an example, Lyotard cited Bourbaki as a group which produces knowledge within a given system of rules.{{sfn|Aubin|pp=332–33}}<ref name="Lyotard">{{cite book |last=Lyotard |first=Jean-François |title=The Postmodern Condition: A Report on Knowledge |publisher=University of Minnesota Press |series=Theory and History of Literature |volume=10 |isbn=978-0816611737 |pages=43, 57–60 |year=1984 }}</ref> Lyotard contrasted Bourbaki's hierarchical, "structuralist" mathematics with the [[catastrophe theory]] of [[René Thom]] and the fractals of [[Benoit Mandelbrot]],{{efn|Mandelbrot was the nephew of Bourbaki founder Szolem Mandelbrojt.{{sfn|Beaulieu|1993|p=31}}<ref>{{cite web |url=http://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/mandelbrot-benoit.pdf |title=Benoit B. Mandelbrot, 1924–2010: A Biographical Memoir by Michael Frame |last=Frame |first=Michael |date=2014 |publisher=[[National Academy of Sciences]] |website=nasonline.org |page=2}}</ref>  Like early Bourbaki associate Gaston Julia, Mandelbrot also worked on fractals.}} expressing preference for the latter "postmodern science" which problematized mathematics with "fracta, catastrophes, and pragmatic paradoxes".{{sfn|Aubin|pp=332–33}}<ref name="Lyotard" />

Although biographer [[Amir Aczel]] stressed Bourbaki's influence on other disciplines during the mid-20th century, Maurice Mashaal moderated the claims of Bourbaki's influence in the following terms:

{{Blockquote|text=While Bourbaki's structures were often mentioned in social science conferences and publications of the era, it seems that they didn't play a real role in the development of these disciplines.  David Aubin, a science historian who analyzed Bourbaki's role in the structuralist movement in France, believes Bourbaki's role was that of a "cultural connector".{{sfn|Aubin|p=297}}  According to Aubin, while Bourbaki didn't have any mission outside of mathematics, the group represented a sort of link between the various cultural movements of the time.  Bourbaki provided a simple and relatively precise definition of concepts and structures, which philosophers and social scientists believed was fundamental within their disciplines and in bridges among different areas of knowledge.  Despite the superficial nature of these links, the various schools of structuralist thinking, including Bourbaki, were able to support each other.  So, it is not a coincidence that these schools suffered a simultaneous decline in the late 1960s. |title=Maurice Mashaal, citing David Aubin{{sfn|Mashaal|p=73}}{{efn|Maurice Mashaal and Amir Aczel each wrote separate biographies on Bourbaki, both published in 2006.  In a review of both books, [[Michael Atiyah]] wrote that "the basic historical facts are well known and are set out in both the books under review".  However Atiyah identified Mashaal's book as the better of the two and criticized Aczel's book, writing: "I was not convinced of the total reliability of its (Aczel's) sources, nor of its philosophical credentials."  Atiyah also wrote that the collaboration between Weil and Lévi-Strauss was a "slightly tenuous link" which Aczel used to make "grand" claims on the scale of Bourbaki's interdisciplinary influence.<ref name="Atiyah">{{cite web |url=https://www.ams.org/notices/200709/tx070901150p.pdf |title=Book Review: Bourbaki, A Secret Society of Mathematicians and The Artist and the Mathematician, Reviewed by Michael Atiyah |last=Atiyah |first=Michael |publisher=American Mathematical Society |website=ams.org}}</ref>}}{{efn|In a 2011 letter to the ''Mathematical Intelligencer'', the mathematician [[:de:Jean-Michel Kantor|Jean-Michel Kantor]]<small>[[:de:Wikipedia:Hauptseite|[de]]]</small> was harshly critical of the notion that Bourbaki's mathematical structures had anything to do with the structuralism of the humanities, rejecting the connections made by Aczel in 2006.<ref name="Kantor">{{cite journal |last=Kantor |first=Jean-Michel |title=Bourbaki's Structures and Structuralism |journal=Mathematical Intelligencer |volume=33 |issue=1 |page=1 |date=2011 |url=https://www.researchgate.net/publication/251293966 |doi=10.1007/s00283-010-9173-4 |doi-access=free }}</ref>  Kantor observed that the two versions of structuralism had developed independently of one another, and that Lévi-Strauss' conception of structure had derived from the [[Prague linguistic circle|Prague circle]] of linguistics, not from Bourbaki.  On the other hand, Aczel had already acknowledged the linguistic origins of the structuralism of the humanities.{{sfn|Aczel|pp=149–59}}  In 1997 David Aubin had pre-emptively moderated both extremes, observing that the two schools of thought had distinct origins, but also had certain interactions and "common features".  Aubin also cited Lévi-Strauss to show that the latter had reached certain conclusions in anthropology independently of Weil's mathematical help, although Weil's help provided confirmation of Lévi-Strauss' conclusions.{{sfn|Aubin|p=311}}  This undermined Aczel's argument that mathematics and Bourbaki played an important role in the development of structuralism in the humanities, although Aubin also stressed that the two schools had some collaboration.}} }}

The impact of "structuralism" on mathematics itself was also criticized. The mathematical historian Leo Corry argued that Bourbaki's use of mathematical structures was unimportant within the ''Éléments'', having been established in ''Theory of Sets'' and cited infrequently afterwards.<ref name="Corry1992">{{cite journal |last=Corry |first=Leo |s2cid=16981077 |author-link=Leo Corry |title=Nicolas Bourbaki and the concept of Mathematical Structure |url=https://www.tau.ac.il/~corry/publications/articles/bourbaki-structures-synthese.html |journal=Synthese |volume=92 |issue=3 |pages=328–32 |date=September 1992 |doi=10.1007/BF00414286 }}</ref><ref name="Corry2001">{{cite book |last=Corry |first=Leo |title=Changing Images in Mathematics: From the French Revolution to the New Millennium |chapter=Mathematical Structures from Hilbert to Bourbaki: The Evolution of an Image of Mathematics |editor-last1=Bottazzini |editor-first1=Umberto |editor-last2=Dalmedico |editor-first2=Amy Dahan |url=https://www.tau.ac.il/~corry/publications/articles/images-structures.html |publisher=Routledge |pages=1–3, 17–18 |date=2001 |isbn=978-0415868273 }}</ref>{{sfn|Corry|2004|p=338}}{{sfn|Corry|2009|pp=25–31}} Corry described the "structural" view of mathematics promoted by Bourbaki as an "image of knowledge"—a conception about a scientific discipline—as opposed to an item in the discipline's "body of knowledge", which refers to the actual scientific results in the discipline itself.<ref name="Corry2001" />

Bourbaki also had some influence in the arts. The literary collective [[Oulipo]] was founded on 24 November 1960 under circumstances similar to Bourbaki's founding, with the members initially meeting in a restaurant. Although several members of Oulipo were mathematicians, the group's purpose was to create [[experimental literature]] by playing with language. Oulipo frequently employed mathematically-based [[constrained writing]] techniques, such as the [[Oulipo#Constraints|S+7 method]]. Oulipo member [[Raymond Queneau]] attended a Bourbaki conference in 1962.{{sfn|Mashaal|p=73}}{{sfn|Aczel|pp=173–82}}

In 2016, an anonymous group of economists collaboratively wrote a note alleging academic misconduct by the authors and editor of a paper published in the ''[[American Economic Review]]''.<ref name="Bearbaki">{{cite web | url=https://mpra.ub.uni-muenchen.de/71699/ | title=A Comment on "Family Ruptures, Stress, and the Mental Health of the Next Generation" | last=Nicolas |first= Bearbaki |date=June 4, 2016 | access-date = February 1, 2021}}</ref><ref>{{cite web | url=https://retractionwatch.com/2016/05/26/economists-go-wild-over-overlooked-citations-in-preprint-on-prenatal-stress/ | date=May 26, 2016| title = Economists go wild over overlooked citations in preprint on prenatal stress|publisher=Retraction Watch|access-date=February 1, 2021}}</ref> The note was published under the name Nicolas Bearbaki in homage to Nicolas Bourbaki.<ref>{{cite web | url=https://statmodeling.stat.columbia.edu/2016/09/23/why-doesnt-this-apparent-case-of-plagiarism-bother-me-at-a-gut-level/ | title= Andrew Gelman is not the plagiarism police because there is no such thing as the plagiarism police. | last=Andrew |first= Gelman |date= September 23, 2016 | access-date = February 1, 2021}}</ref>

In 2018, the American musical duo [[Twenty One Pilots]] released a [[concept album]] named ''[[Trench (album)|Trench]]''. The album's conceptual framework was the mythical city of "Dema" ruled by nine "bishops"; one of the bishops was named "Nico", short for Nicolas Bourbaki.  Another of the bishops was named Andre, which may refer to André Weil.  Following the album's release, there was a spike in internet searches for "Nicolas Bourbaki".<ref name="numericana" />{{efn|Similarly, Bourbaki created nicknames for its members.  Jean Delsarte was referred to as "bishop", which may have been a reference to his Catholicism.{{sfn|Mashaal|p=111}} }}

===Praise===

Bourbaki's work has been praised by some mathematicians. In a book review, [[Emil Artin]] described the ''Éléments'' in broad, positive terms:

{{blockquote|Our time is witnessing the creation of a monumental work: an exposition of the whole of present day mathematics. Moreover this exposition is done in such a way that the common bond between the various branches of mathematics become clearly visible, that the framework which supports the whole structure is not apt to become obsolete in a very short time, and that it can easily absorb new ideas.|Emil Artin<ref name="Artin" /> }}

Among the volumes of the ''Éléments'', Bourbaki's work on Lie Groups and Lie Algebras has been identified as "excellent",<ref name="Atiyah" /> having become a standard reference on the topic. In particular, former member Armand Borel described the volume with chapters 4–6 as "one of the most successful books by Bourbaki".{{sfn|Borel|p=379}} The success of this part of the work has been attributed to the fact that the books were composed while leading experts on the topic were Bourbaki members.{{sfn|Senechal|pp=22–28}}{{sfn|Aczel|p=111}}

[[Jean-Pierre Bourguignon]] expressed appreciation for the Séminaire Bourbaki, saying that he'd learned a large amount of material at its lectures, and referred to its printed lecture notes regularly.{{sfn|Mashaal|p=102}} He also praised the ''Éléments'' for containing "some superb and very clever proofs".{{sfn|Mashaal|pp=54–55}}

===Criticism===

Bourbaki has also been criticized by several mathematicians—including its own former members—for a variety of reasons. Criticisms have included the choice of presentation of certain topics within the ''Éléments'' at the expense of others,{{efn|This specific point has itself been criticized.  It has been observed that it is unfair to criticize a work on a given topic for not dealing with other topics.<ref>{{cite web |url=http://www.math.nsc.ru/LBRT/g2/english/ssk/euclid.html |title=Apology of Euclid |author-link=Semën Samsonovich Kutateladze |last=Kutateladze |first=Semën Samsonovich }}</ref>{{sfn|Mashaal|pp=116–18}} }} dislike of the method of presentation for given topics, dislike of the group's working style, and a perceived [[Elitism|elitist]] mentality around Bourbaki's project and its books, especially during the collective's most productive years in the 1950s and 1960s.

Bourbaki's deliberations on the ''Éléments'' resulted in the inclusion of some topics, while others were not treated. When asked in a 1997 interview about topics left out of the ''Éléments'', former member Pierre Cartier replied:

{{blockquote|text=There is essentially no analysis beyond the foundations: nothing about [[partial differential equations]], nothing about [[probability]]. There is also nothing about [[combinatorics]], nothing about [[algebraic topology]],{{efn|Bourbaki has since published a book on algebraic topology.}} nothing about concrete [[geometry]]. And Bourbaki never seriously considered [[mathematical logic|logic]]. Dieudonné himself was very vocal against logic.  Anything connected with [[mathematical physics]] is totally absent from Bourbaki's text.|title=Pierre Cartier{{sfn|Senechal|pp=22–28}} }}

Although Bourbaki had resolved to treat mathematics from its foundations, the group's eventual solution in terms of set theory was attended by several problems. Bourbaki's members were mathematicians as opposed to [[logician]]s, and therefore the collective had a limited interest in [[mathematical logic]].{{sfn|Guedj|p=20}} As Bourbaki's members themselves said of the book on set theory, it was written "with pain and without pleasure, but we had to do it."{{sfn|Mashaal|p=121}} Dieudonné personally remarked elsewhere that ninety-five percent of mathematicians "don't care a fig" for mathematical logic.{{sfn|Mashaal|p=120}} In response, logician Adrian Mathias harshly criticized Bourbaki's foundational framework, noting that it did not take [[Gödel]]'s results into account.{{sfn|Mashaal|pp=120–23}}<ref name="Mathias">{{Cite web|url=https://www.dpmms.cam.ac.uk/~ardm/bourbaki.pdf|title=The Ignorance of Bourbaki|last=Mathias|first=Adrian|date=August 22, 1990|website=dpmms.cam.ac.uk}}</ref>

Bourbaki also influenced the New Math, a failed{{sfn|Mashaal|p=135}} reform in Western mathematics education at the elementary and secondary levels, which stressed abstraction over concrete examples. During the mid-20th century, reform in basic math education was spurred by a perceived need to create a mathematically literate workforce for the modern economy, and also to compete with the [[Soviet Union]]. In France, this led to the Lichnerowicz Commission of 1967, headed by [[André Lichnerowicz]] and including some (then-current and former) Bourbaki members. Although Bourbaki members had previously (and individually) reformed math instruction at the university level, they had less direct involvement with implementation of the New Math at the primary and secondary levels. New Math reforms resulted in instructional material which was incomprehensible to both students and teachers, failing to meet the [[Cognition|cognitive]] needs of younger students. The attempted reform was harshly criticized by Dieudonné and also by brief founding Bourbaki participant Jean Leray.{{sfn|Mashaal|pp=134–45}} Apart from French mathematicians, the French reforms also met with harsh criticism from Soviet-born mathematician [[Vladimir Arnold]], who argued that in his time as a student and teacher in Moscow, the teaching of mathematics was firmly rooted in analysis and geometry, and interweaved with problems from classical mechanics; hence, the French reforms cannot be a legitimate attempt to emulate Soviet scientific education. In 1997, while speaking to a conference on mathematical teaching in Paris, he commented on Bourbaki by stating: "genuine mathematicians do not gang up, but the weak need gangs in order to survive." and suggested that Bourbaki's bonding over "super-abstractness" was similar to groups of mathematicians in the 19th century who had bonded over anti-Semitism.<ref>{{Cite web|url=https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html|title = V.I. Arnold, on teaching mathematics}}</ref>

[[File:Benoit_Mandelbrot_mg_1804c.jpg|thumb|left|200px|[[Benoit Mandelbrot]] was among Bourbaki's critics]]

Dieudonné later regretted that Bourbaki's success had contributed to a [[snob]]bery for pure mathematics in France, at the expense of [[applied mathematics]]. In an interview, he said: "It is possible to say that there was no serious applied mathematics in France for forty years after Poincaré. There was even a snobbery for pure math. When one noticed a talented student, one would tell him 'You should do pure math.' On the other hand, one would advise a mediocre student to do applied math while thinking, "It's all that he can do! ... The truth is actually the reverse. You can't do good work in applied math until you can do good work in pure math."{{sfn|Mashaal|pp=118–19}} Claude Chevalley confirmed an elitist culture within Bourbaki, describing it as "an absolute certainty of our superiority over other mathematicians."{{sfn|Guedj|p=20}} Alexander Grothendieck also confirmed an elitist mentality within Bourbaki.{{sfn|Corry|2009|pp=38–51}} Some mathematicians, especially geometers and applied mathematicians, found Bourbaki's influence to be stifling.{{sfn|Aubin|p=313}} Benoit Mandelbrot's decision to emigrate to the United States in 1958 was motivated in part by a desire to escape Bourbaki's influence in France.{{sfn|Mashaal|p=130}}

Several related criticisms of the ''Éléments'' have concerned its target audience and the intent of its presentation. Volumes of the ''Éléments'' begin with a note to the reader which says that the series "takes up mathematics at the beginning, and gives complete proofs" and that "the method of exposition we have chosen is axiomatic and abstract, and normally proceeds from the general to the particular."<ref>''Theory of Sets'', p. v.</ref> Despite the opening language, Bourbaki's intended audience are not absolute beginners in mathematics, but rather undergraduates, graduate students, and professors who are familiar with mathematical concepts.{{sfn|Mashaal|p=54}} Claude Chevalley said that the ''Éléments'' are "useless for a beginner",{{sfn|Guedj|p=22}} and Pierre Cartier clarified that "The misunderstanding was that it should be a textbook for everybody. That was the big disaster."{{sfn|Senechal|pp=22–28}}

The work is divided into two halves. While the first half—the ''Structures fondamentales de l’analyse''—treats established subjects, the second half deals with modern research areas like commutative algebra and spectral theory. This divide in the work is related to a historical change in the intent of the treatise. The ''Éléments'<nowiki/>'' content consists of theorems, proofs, exercises and related commentary, common material in math textbooks. Despite this presentation, the first half was not written as [[original research]] but rather as a reorganized presentation of established knowledge. In this sense, the ''Éléments''' first half was more akin to an [[encyclopedia]] than a textbook series. As Cartier remarked, "The misunderstanding was that many people thought it should be ''taught'' the way it was written in the books. You can think of the first books of Bourbaki as an encyclopedia of mathematics... If you consider it as a textbook, it's a disaster."{{sfn|Senechal|pp=22–28}}

The strict, ordered presentation of material in the ''Éléments''' first half was meant to form the basis for any further additions. However, developments in modern mathematical research have proven difficult to adapt in terms of Bourbaki's organizational scheme. This difficulty has been attributed to the fluid, dynamic nature of ongoing research which, being new, is not settled or fully understood.<ref name="Atiyah" />{{sfn|Borel|pp=377–379}} Bourbaki's style has been described as a particular scientific [[paradigm]] which has been superseded in a [[paradigm shift]]. For example, [[Ian Stewart (mathematician)|Ian Stewart]] cited [[Vaughan Jones|Vaughan Jones']] novel work in [[knot theory]] as an example of topology which was done without dependence on Bourbaki's system.<ref name="Stewart">{{Cite journal |last=Stewart |first=Ian |title=Bye-Bye Bourbaki: Paradigm Shifts in Mathematics |journal=The Mathematical Gazette |date=November 1995 |volume=79 |pages=496–98 |publisher=[[The Mathematical Association]] |doi=10.2307/3618076 |issue=486 |jstor=3618076|s2cid=125418650 }}</ref> Bourbaki's influence has declined over time;<ref name="Stewart" /> this decline has been partly attributed to the absence of certain modern topics—such as category theory—from the treatise.{{sfn|Aczel|p=205}}{{sfn|Mashaal|pp=81–84}}

Although multiple criticisms have pointed to shortcomings in the collective's project, one has also pointed to its strength: Bourbaki was a "victim of its own success"<ref name="Atiyah" /> in the sense that it accomplished what it set out to do, achieving its original goal of presenting a thorough treatise on modern mathematics.{{sfn|Aczel|pp=204–05}}{{sfn|Aubin|p=329}}{{sfn|Borel|p=377}} These factors prompted biographer Maurice Mashaal to conclude his treatment of Bourbaki in the following terms:

{{blockquote|text=Such an enterprise deserves admiration for its breadth, for its enthusiasm and selflessness, for its strongly collective character.  Despite some mistakes, Bourbaki did add a little to 'the honor of the human spirit'.  In an era when sports and money are such great idols of civilization, this is no small virtue.|title=Maurice Mashaal{{sfn|Mashaal|p=153}} }}

==See also==
* [[Bourbaki–Witt theorem]]
* [[Jacobson–Bourbaki theorem]]
* [[Secret society]]

'''Other collective mathematical pseudonyms'''
* [[Arthur Besse]]
* [[Blanche Descartes]]
* [[John Rainwater]]
* [[G. W. Peck]]

==Notes==
{{notelist}}

==References==
{{reflist|colwidth=30em}}

==Bibliography==
* {{cite book |last=Aczel |first=Amir D. |author-link=Amir Aczel |title=The Artist and the Mathematician: the Story of Nicolas Bourbaki, the Genius Mathematician Who Never Existed |year=2006 |publisher=Thunder's Mouth Press |isbn=978-1560259312 |ref={{sfnRef|Aczel}} }}
* {{cite journal |last=Aubin |first=David |date=1997 |title=The Withering Immortality of Nicolas Bourbaki: A Cultural Connector at the Confluence of Mathematics, Structuralism, and the Oulipo in France |journal=Science in Context |publisher=[[Cambridge University Press]] |volume=10 |issue=2 |pages=297–342 |url=https://webusers.imj-prg.fr/~david.aubin/publis/1997.pdf |doi=10.1017/S0269889700002660 |s2cid=170683589 |ref={{sfnRef|Aubin}} }}
* {{cite journal |last=Beaulieu |first=Liliane |date=1993 |title=A Parisian Café and Ten Proto-Bourbaki Meetings (1934–1935) |url=https://www.yumpu.com/en/document/read/4320353/karen-v-h-parshall-a-parisian-cafe-and-ten-proto-bourbaki- |journal=[[The Mathematical Intelligencer]] |volume=15 |issue=1 |pages=27–35 |doi=10.1007/BF03025255 |s2cid=189888171 }}
* {{cite journal |last=Beaulieu |first=Liliane |date=1999 |title=Bourbaki's Art of Memory |url=http://dodo.pdmi.ras.ru/~pavlov/bourbaki.pdf |journal=[[Osiris (journal)|Osiris]] |volume=14 |pages=219–51 |doi=10.1086/649309 |s2cid=143559711 }}
* {{cite journal |last=Borel |first=Armand |author-link=Armand Borel |date=March 1998 |title=Twenty-Five Years with Nicolas Bourbaki, (1949–1973) |url=https://www.ams.org/notices/199803/borel.pdf |journal=[[Notices of the American Mathematical Society]] |volume=45 |issue=3 |pages=373–80 |ref={{sfnRef|Borel}} }}
* {{cite journal |last=Bourbaki |first=Nicolas |title=The Architecture of Mathematics |journal=[[American Mathematical Monthly]] |volume=57 |issue=4 |pages=221–32 |date=1950 |jstor=2305937 |doi=10.1080/00029890.1950.11999523 |ref={{sfnRef|Bourbaki 1950}} }} Presumptive author: Jean Dieudonné. Authorized translation of the book chapter ''L'architecture des mathématiques'', appearing in English as a journal article.
* {{cite book |last=Corry |first=Leo |author-link=Leo Corry |title=Modern Algebra and the Rise of Mathematical Structures |chapter=Nicolas Bourbaki: Theory of Structures |url=https://www.tau.ac.il/~corry/publications/articles/bourbaki-structures.html |publisher=Springer |pages=289–338 |date=2004 |isbn=978-3764370022 }}
* {{cite book |last=Corry |first=Leo |title=The Oxford Handbook of the History of Mathematics |chapter=Writing the Ultimate Mathematical Textbook: Nicolas Bourbaki's ''Éléments de mathématique'' |editor-last1=Robson |editor-first1=Eleanor |editor-last2=Stedall |editor-first2=Jacqueline |url=https://www.tau.ac.il/~corry/publications/articles/Bourbaki%20-%20OHHM.html |publisher=[[Oxford University Press]] |pages=565–87 |date=2009 |isbn=978-0199213122 }}
* {{cite journal |last=Guedj |first=Denis |author-link=Denis Guedj |translator-last=Gray |translator-first=Jeremy |translator-link=Jeremy Gray |date=1985 |title=Nicholas Bourbaki, Collective Mathematician : an Interview with Claude Chevalley |url=https://www.ocf.berkeley.edu/~lekheng/interviews/ClaudeChevalley.pdf |journal=Mathematical Intelligencer |volume=7 |issue=2 |pages=18–22 |doi=10.1007/BF03024169 |s2cid=123548747 |ref={{sfnRef|Guedj}} }}
*{{cite journal|last=Krömer|first=Ralf|title=La "machine de Grothendieck" se fonde-t-elle seulement sur des vocables métamathématiques? Bourbaki et les catégories au cours des années cinquante|journal=Revue d'histoire des mathématiques|volume=12|year=2006|pages=119–162|url=http://www.numdam.org/item/RHM_2006__12_1_119_0/|doi=10.24033/rhm.38|zbl=1177.01034}}
* {{cite book |last=Mashaal |first=Maurice |title=Bourbaki: a Secret Society of Mathematicians |year=2006 |publisher=[[American Mathematical Society]] |isbn=978-0821839676 |ref={{sfnRef|Mashaal}} }}
* {{cite journal |last=Senechal |first=Marjorie |author-link=Marjorie Senechal |title=The Continuing Silence of Bourbaki: an Interview with Pierre Cartier, June 18, 1997 |journal=Mathematical Intelligencer |year=1998 |volume=20 |pages=22–28 |url=http://www.ega-math.narod.ru/Bbaki/Cartier.htm |doi=10.1007/BF03024395 |s2cid=124159858 |ref={{sfnRef|Senechal}} }}

==External links==
* [http://www.bourbaki.fr/ Official Website of ''L'Association des Collaborateurs de Nicolas Bourbaki''] {{in lang|fr}}
* [http://sites.mathdoc.fr/archives-bourbaki/ Archives of the association] {{in lang|fr}}

{{Authority control}}

{{DEFAULTSORT:Bourbaki, Nicolas}}
[[Category:Nicolas Bourbaki| ]]
[[Category:1934 establishments in France]]
[[Category:1935 establishments in France]]
[[Category:Academic shared pseudonyms]]
[[Category:French mathematicians]]
[[Category:Large-scale mathematical formalization projects]]
[[Category:Organizations established in 1934]]
[[Category:Organizations established in 1935]]
[[Category:Pseudonymous mathematicians]]
[[Category:Secret societies in France]]
[[Category:Collaborative non-fiction]]